Proof. These statements are conjectured in Li-nv?, and proved in Blu-existence?, LX-higher-rank?, Xu-quasimonomial?, XZ-uniqueness?, XZ-SDC?. See also BLQ-convexity?, Che-HRFG? for different proofs of certain parts of the statements. ◻

Proof. See LX-Kol-comp-stab? or XZ-uniqueness?. ◻

Proof. Since \(N(K_X+\Delta)\) is Cartier at \(x\), after replacing \(X\) with a smaller open neighborhood of \(x\) we may assume that \(N(K_X+\Delta)\sim 0\). This does not change \(\mathscr{X}_0\), hence does not affect the statement we want to prove. Since \(\mathscr{X}\setminus \mathscr{X}_0 \cong X\times (\mathbb{A}^1\setminus\{0\})\), on the total space \(\mathscr{X}\) we get a Weil divisor \(L:=N(K_{\mathscr{X}}+\Delta_{\mathscr{X}})\) such that \(L|_{\mathscr{X}\setminus \mathscr{X}_0} \sim 0\). Since \(\mathscr{X}_0\) is integral and \(\mathscr{X}_0\sim 0\), the natural restriction \(\mathrm{Cl}(\mathscr{X})\to \mathrm{Cl}(\mathscr{X}\setminus \mathscr{X}_0)\) is an isomorphism. It follows that \(L\sim 0\). In particular, \(N(K_{\mathscr{X}}+\Delta_{\mathscr{X}})\) is Cartier at \(x_0\). ◻

Proof. This is well known to the experts. We include a proof for the reader’s convenience. All we need to prove is that \(\mathscr{X}_0\) and \(\mathscr{G}\) give the same valuation of \(\mathbb{C}(X\times \mathbb{A}^1)\). Since both divisors are invariant under the \(\mathbb{G}_m\)-action, we only need to check that every \(\mathbb{G}_m\)-equivariant function has the same vanishing order along \(\mathscr{X}_0\) and \(\mathscr{G}\). In other words, it suffices to show that \(\mathrm{ord}_{\mathscr{X}_0}(a\cdot t^m)=\mathrm{ord}_{\mathscr{G}}(a\cdot t^m)\) for all \(a\in R\) and \(m\in\mathbb{N}\). Let \(l=\mathrm{ord}_D(a)\). Then by definition \(\mathrm{ord}_{\mathscr{G}}(a\cdot t^m) = l+m\). On the other hand, for any \(s\in R[t]\), \(\mathrm{ord}_{\mathscr{X}_0}(s)\) is (by definition) the largest integer \(r\) such that \(t^{-r}s\in \mathcal{R}\), thus from 3 we get \(\mathrm{ord}_{\mathscr{X}_0}(a\cdot t^m) = l+m\) as well. ◻

Proof. This mostly follows from LX-Kol-comp-stab? or XZ-SDC?, but we also give a direct proof for the reader’s convenience. As \(-D\) is \(\pi\)-ample, we have \(Y = \mathrm{Proj}_X \left( \bigoplus_{m\in \mathbb{N}} \mathfrak{a}_m \right)\) where \(\mathfrak{a}_m := \mathfrak{a}_m(\mathrm{ord}_D) = \pi_* \mathcal{O}_Y(-mD)\) as before, and \[D\cong \mathrm{Proj}(\bigoplus_{m\in\mathbb{N}} \mathfrak{a}_m/\mathfrak{a}_{m+1})=\mathrm{Proj}(\mathrm{gr}_D R)\] by LZ-Tian-sharpness?. The flat extension \(\widetilde{\mathfrak{a}}_m\) of \(\mathfrak{a}_m\) to \(\mathscr{X}\) is given by \[\widetilde{\mathfrak{a}}_m = \mathcal{R}\cap \mathfrak{a}_m[t,t^{-1}] = \bigoplus_{l\in\mathbb{Z}} t^{-l} \mathfrak{a}_{\max\{l,m\}} \subseteq \mathcal{R}.\] Finite generation of \(\mathcal{R}\) implies that \(\bigoplus_{m\in \mathbb{N}} \widetilde{\mathfrak{a}}_m\cong \bigoplus_{m\in\mathbb{N},\,l\le m} t^{-l}\mathfrak{a}_m\) is also finitely generated. Let \(\mathscr{Y}:=\mathrm{Proj}_{\mathscr{X}} \left( \bigoplus_{m\in \mathbb{N}} \widetilde{\mathfrak{a}}_m \right)\) with induced map \(\psi\colon \mathscr{Y}\to \mathscr{X}\). Then (1) is satisfied by construction. Note that \(\widetilde{\mathfrak{a}}_m\) is invariant under the \(\mathbb{G}_m\)-action, hence the \(\mathbb{G}_m\)-action lifts to \(\mathscr{Y}\). We have \[\widetilde{\mathfrak{a}}_m/\widetilde{\mathfrak{a}}_{m+1} \cong \bigoplus_{l\in \mathbb{Z},\,l\le m} t^{-l}\cdot (\mathfrak{a}_m/\mathfrak{a}_{m+1}),\] thus the \(\psi\)-exceptional locus \(\mathscr{D}\) is \(\mathrm{Proj}(\bigoplus_{m\in \mathbb{N}} \widetilde{\mathfrak{a}}_m/\widetilde{\mathfrak{a}}_{m+1})\cong \mathrm{Proj}((\mathrm{gr}_D R)[t])\cong D\times \mathbb{A}^1\). Under this isomorphism, the induced \(\mathbb{G}_m\)-action on \(\mathscr{D}\) is trivial on the factor \(D\). It is also not hard to see that \(\Delta_{\mathscr{D}}\) does not have any vertical component over \(\mathbb{A}^1\) (more generally, for any \(\mathbb{G}_m\)-invariant Cartier divisor \(H=(f=0)\) on \(X\), where \(0\neq f\in R\), the intersection \(\psi_*^{-1}H_{\mathscr{X}}\cap \mathscr{D}\) does not have any vertical component; this can be checked using the explicit presentation of \(\mathscr{Y}\) and \(\mathscr{D}\) above). Thus as \(\Delta_{\mathscr{D}}\) is \(\mathbb{G}_m\)-invariant, we see that \((\mathscr{D}, \Delta_{\mathscr{D}})\) is a trivial family over \(\mathbb{A}^1\). It follows that \((\mathscr{D}, \Delta_{\mathscr{D}})\cong (D, \Delta_D) \times \mathbb{A}^1\) as its fiber over \(0\neq t\in \mathbb{A}^1\) is \((D,\Delta_D)\) by (1). ◻

Proof. We largely follow the proof of XZ-uniqueness?. By the discussion in Paragraph 18, the Reeb vector \(\xi\) generates a \(\mathbb{G}_m\)-action on \((X,\Delta)\), the Seifert \(\mathbb{G}_m\)-bundle \(\varphi\colon X\setminus\{x\}\to D\) is given by some \(\mathbb{Q}\)-divisor \(L\) on \(D\) and we have \[R=\bigoplus_{m\in\mathbb{N}} H^0(D,mL).\] Note that the torus \(\mathbb{T}\) also acts on the log Fano pair \((D,\Delta_D)\). Fix a positive integer \(r\) such that \(rL\) is Cartier. For any quasi-monomial valuation \(w\in \mathrm{Val}_D\) and any \(t\ge 0\), let \(w_t\in \mathrm{Val}_{X,x}\) be the quasi-monomial valuation on \(X\) defined by \[\begin{align} \label{eq-wt} w_t(s) = w(s)+ t\cdot \mathrm{wt}_\xi(s) \end{align}\tag{8}\] for any \(m\in\mathbb{N}\) and any \(0\neq s\in H^0(D,mL)\), where we set \(w(s):=\frac{1}{r}w(s^r)\). Note that \(w_t\in \mathrm{Val}_{X,x}\) if \(t>0\) and it is \(\mathbb{T}\)-invariant if and only if \(w\) is \(\mathbb{T}\)-invariant. We claim that \[\label{eq-compare32A} A_{X,\Delta}(w_t) = A_{D,\Delta_D}(w)+t\cdot A_{X,\Delta}(\xi)\, .\tag{9}\] Indeed, if \(L\) is Cartier, then \(A_{X,\Delta}(w_0)=A_{D,\Delta_D}(w)\) as \((X,\Delta)\) is a cone over \((D,\Delta_D)\), and 9 follows from the observation that \(w_t\) is a monomial combination of \(w_0\) and \(\mathrm{wt}_\xi\) (i.e. on some log smooth model \((Y,E)\) of \((X,\Delta)\) the valuations \(w_t\)’s and \(\mathrm{wt}_\xi\) belong to the same simplex of \(\mathrm{QM}(Y,E)\) and the equality \(w_t=w_0+t\cdot \mathrm{wt}_\xi\) holds in this simplex) and the fact that the log discrepancy function is linear on each simplex of \(\mathrm{QM}(Y,E)\). In the general case, using the \(\mu_r\subseteq \mathbb{G}_m\) action on \(x\in (X,\Delta)\) we let \[\big(x'\in (X',\Delta')\big) = \big(x\in (X,\Delta)\big)/\mu_r,\] i.e. \(X'=X/\mu_r\), and if \(f\colon X\to X'\) is the induced map, then \(x'=f(x)\) and \[K_X+\Delta = f^*(K_{X'}+\Delta').\] Let \(\xi'\) be the induced Reeb vector on \(X'\) so that \(\mathrm{wt}_{\xi'}\in \mathrm{Val}_{X'}\) is the restriction of \(\mathrm{wt}_\xi\), and let \(w'_t\in \mathrm{Val}_{X'}\) be the restriction of \(w_t\). Note that \(X'=\mathrm{Spec}(R')\) where \(R':=\bigoplus_{m\in\mathbb{N}} H^0(D,mrL)\), hence as \(rL\) is Cartier and combined with 5 we know that \((X',\Delta')\) is a cone over \((D,\Delta_D)\). By construction, \(w'_t(s) = w(s)+ t\cdot \mathrm{wt}_{\xi'}(s)\) for all \(s\in H^0(D,mrL)\), thus from the special case of cones treated above we deduce that \[A_{X',\Delta'}(w'_t) = A_{D,\Delta_D}(w)+t\cdot A_{X',\Delta'}(\xi').\] By KM98?, we also have \(A_{X,\Delta}(w_t)=A_{X',\Delta'}(w'_t)\) and \(A_{X,\Delta}(\xi)=A_{X',\Delta'}(\xi')\). This gives 9 .

On the other hand, by XZ-uniqueness?, we have \(S(\xi;w_0) = S_{D,\Delta_D}(w)\). By 8 , we also see that \(\widetilde{S}(\xi;w_t) = \widetilde{S}(\xi;w_0)+t\cdot \widetilde{S}(\xi;\xi)\), hence \[\label{eq-compare32S} S(\xi;w_t) = S_{D,\Delta_D}(w) + t\cdot A_{X,\Delta}(\xi)\, .\tag{10}\] Therefore, if \(\delta(D,\Delta_D)\ge 1-\varepsilon\), then by combining Equations 9 and 10 , we have \[\begin{align} A_{X,\Delta}(w_t)&=&A_{D,\Delta_D}(w)+t\cdot A_{X,\Delta}(\xi) \\ &\ge& (1-\varepsilon)\cdot S_{D,\Delta_D}(w)+t\cdot A_{X,\Delta}(\xi) \\ &\ge &(1-\varepsilon)S(\xi;w_t) \end{align}\] for any quasi-monomial valuation \(w\in \mathrm{Val}_D\) and any \(t>0\), where the first inequality is Xu-book?. Since \(A_{X,\Delta}(\xi) = S(\xi;\xi)\) by definition, and every \(\mathbb{T}\)-invariant quasi-monomial valuation in \(\mathrm{Val}_{X}\) other than \(\lambda\cdot \mathrm{wt}_\xi\) is of the form \(w_t\), we conclude that ?? holds for all \(\mathbb{T}\)-invariant quasi-monomial valuations \(v\in \mathrm{Val}_{X,x}\).

Conversely, if \(A_{X,\Delta}(w_t) \ge (1-\varepsilon)S(\xi;w_t)\) holds for all \(\mathbb{T}\)-invariant quasi-monomial valuations \(v\in \mathrm{Val}_{X,x}\), then letting \(t\to 0^+\) we get (again by 9 and 10 ) that \[A_{D,\Delta_D}(w)\ge (1-\varepsilon)\cdot S_{D,\Delta_D}(w)\] for all \(\mathbb{T}\)-invariant quasi-monomial valuations \(w\in \mathrm{Val}_D\). By Z-equivariant-K?, this implies \(\delta(D,\Delta_D)\ge 1-\varepsilon\). ◻

Proof. Let \(\mathbb{T}\) be the torus acting on \(x\in (X,\Delta)\). By Theorem 20, we have \(A_{X,\Delta}(v)\ge S(\xi;v)\) for any \(\mathbb{T}\)-invariant quasi-monomial valuation \(v\in \mathrm{Val}_{X,x}\). We claim that there exists an open neighborhood \(U\subseteq \mathfrak{t}_\mathbb{R}^+\) of \(\xi\) such that \[\label{eq:S32perturbed} S(\xi;v)\ge (1-\varepsilon)S(\eta;v)\tag{11}\] for any \(\eta\in U\) and any \(\mathbb{T}\)-invariant quasi-monomial valuation \(v\in \mathrm{Val}_{X,x}\). The lemma follows immediately from Lemma 23 and this claim.

Since the \(\mathbb{T}\)-action is good, its weight cone \(\sigma\subseteq M_\mathbb{R}\) (generated by those \(\alpha\in M(\mathbb{T})\) with \(R_\alpha\neq 0\)) is strongly convex, thus we can pick some \(0<t\ll 1\) and some open neighborhood \(U\subseteq \mathfrak{t}_\mathbb{R}^+\) of \(\xi\) such that \((1-t)\langle \alpha, \xi\rangle\le \langle \alpha, \eta\rangle\) for any \(\alpha\in \sigma\) and \(\eta\in U\). From the definition of the \(\widetilde{S}_m\) and \(\widetilde{S}\)-invariant, this gives \[\widetilde{S}_m ((1-t)\xi;v)\ge \widetilde{S}_m(\eta;v) \, .\] Hence, for any \(\mathbb{T}\)-invariant quasi-monomial valuation \(v\in \mathrm{Val}_{X,x}\), we get \[\widetilde{S}(\xi;v) = (1-t)^{n+1}\widetilde{S}((1-t)\xi;v)\ge (1-t)^{n+1}\widetilde{S}(\eta;v)\] by 6 . By the first displayed formula on XZ-uniqueness?, we have \(\widetilde{S}(\eta;\eta) = n\cdot \mathrm{vol}(\mathrm{wt}_\eta)\) for any \(\eta\in \mathfrak{t}^+_\mathbb{R}\) (see e.g. ELS03? for the definition of the volume of a valuation). Since both the log discrepancy \(\eta\mapsto A_{X,\Delta}(\eta)\) and the volume function \(\eta\mapsto \mathrm{vol}(\eta)\) are continuous on the Reeb cone by LX-higher-rank?, after possibly shrinking \(U\), we may assume that \[A_{X,\Delta}(\xi)\ge (1-t)A_{X,\Delta}(\eta)\quad \mathrm{and}\quad \widetilde{S}(\xi;\xi)\le (1-t)^{-1} \widetilde{S}(\eta;\eta)\] for any \(\eta\in U\). Putting these estimates together we obtain \[S(\xi;v) \ge (1-t)^{n+3} S(\eta;v),\] thus for sufficiently small \(t\) this yields 11 . ◻

Proof of Theorem 22. Let \(x_0\in (X_0,\Delta_0;\xi_v)\) be the K-semistable log Fano cone degeneration of \(x\in (X,\Delta)\) given by Theorem 6. Since \(\mathrm{gr}_{v} R\) is finitely generated, by LX-higher-rank? we know that there exists an open neighborhood \(U\subseteq \mathrm{QM}(Y,E)\) of \(v\) such that \(\mathrm{gr}_w R\cong \mathrm{gr}_{v} R\) for every quasi-monomial valuation \(w\in U\). The grading on \(\mathrm{gr}_w R\) induces a Reeb vector \(\xi_w\) on \(X_0=\mathrm{Spec}(\mathrm{gr}_v R)\) through this isomorphism. Moreover, \(w\in U\) is a divisorial valuation if and only if \(\xi_w\) is quasi-regular.

The assignment \(w\mapsto \xi_w\) then identifies \(U\) with an open neighborhood of \(\xi_v\) in the Reeb cone \(\mathfrak{t}_\mathbb{R}^+\) of \(X_0\). By Lemma 24, the theorem holds for some open neighborhood \(U_0\) of \(\xi_v\) in the Reeb cone. Shrinking \(U\) if necessary, we may assume that \(\xi_w\in U_0\) for all \(w\in U\). We may further assume by LX-higher-rank? that every divisorial valuation \(w\in U\) is a Kollár component and the corresponding test configuration degenerates \(\Delta\) to \(\Delta_0\). The theorem now follows as the log Fano pairs on the Kollár components corresponding to \(w\) and \(\xi_w\) are isomorphic by Lemma 15. ◻

Proof. We follow the notation of Definition 27. For (1), note that if \(\mathrm{d}\sigma=0\) then \(\mathrm{d}\sigma'=0\) (it suffices to check this over \(\mathscr{X}\setminus \mathscr{X}_0\cong X\times (\mathbb{A}^1\setminus \{0\})\), since the sheaf in question is torsion-free). Thus \(\mathrm{d}(t^m\sigma') = mt^{m-1}\mathrm{d}t\wedge\sigma'=0\) in \(\Omega_{\mathscr{X}/\mathbb{A}^1}^{[p+1]}\) (again it is enough to verify this equality over \(\mathscr{X}\setminus \mathscr{X}_0\)). Restricting to the special fiber gives \(\mathrm{d}\sigma_0 = 0\).

For (2), the forward direction is clear. Suppose that \(\sigma_0^{n/p}\neq 0\). Then since \(\sigma\) is non-degenerate, the zero locus of \(\sigma_{\mathscr{X}}^{n/p} \in H^0(\mathscr{X},\Omega_{\mathscr{X}/\mathbb{A}^1}^{[n]})\) has codimension at least two in \(\mathscr{X}\). Since \(\Omega_{\mathscr{X}/\mathbb{A}^1}^{[n]}\) is reflexive of rank one, this implies that \(\Omega_{\mathscr{X}/\mathbb{A}^1}^{[n]}\cong \mathcal{O}_{\mathscr{X}}\) and \(\sigma_{\mathscr{X}}^{n/p}\) extends to a nowhere vanishing section of \(\Omega_{\mathscr{X}/\mathbb{A}^1}^{[n]}\). In particular, its restriction yields a nowhere vanishing \(n\)-form on \(\mathscr{X}_0^{\mathrm{sm}}\), i.e.\(\sigma_{0}\) is non-degenerate. ◻

Proof. Let \((Y',E')\) and \((Y,E)\) be two log smooth models such that \[v\in \mathrm{QM}(Y,E) \cap \mathrm{QM}(Y',E').\] By taking a common log resolution, we may assume there is a morphism \(\varphi\colon Y'\to Y\) and \(\mathrm{Supp}(\varphi^*E+\mathrm{Ex}(\varphi))\subseteq E'\). In particular, there is a natural map \[\label{eq:pullback32log32differential} \varphi^*(\Omega_Y(\log E))\to \Omega_{Y'}(\log E').\tag{13}\] Suppose the center \(c_{Y'}(v)\) of \(v\) on \(Y'\) is contained in the intersection of components \(E'_1,\ldots, E'_r\subseteq E'\), and let \(a_i:=v(E'_i)>0\) be the coordinate of \(v\) along \(E'_i\) for \(i=1,\ldots,r\). Since \(A_{Y,E}(E'_i)\ge 0\) and \[0=A_{Y,E}(v)=\sum_{i=1}^r a_i A_{Y,E}(E'_i)\] by definition, we have \(A_{Y,E}(E'_i)=0\) for all \(i\), hence as \(\mathrm{Supp}(\varphi^*E+\mathrm{Ex}(\varphi))\subseteq E'\) we get \(\varphi^*(K_Y+E)=K_{Y'}+E'\) near \(c_{Y'}(v)\). This implies that 13 is an isomorphism near \(c_{Y'}(v)\). Therefore, \(v(\pi^*\sigma) = v(\varphi^*\pi^*\sigma)\) and the lemma follows. ◻

Proof. Since the question is local, we may assume that \(X\) is affine and \(\mathcal{F}\) is free with basis \(\{s_i\}\). Since \(s\) is a rational section, there exists some \(a\in H^0(\mathcal{O}_X)\) such that \(a s\in H^0(\mathcal{F})\). Write \(as=\sum_i a_i s_i\) and let \(\mathcal{I}\subseteq \mathcal{O}_X\) be the ideal generated by the \(a_i\)’s so that \(Z=V(\mathcal{I})\subseteq X\) is the zero scheme of \((as)\). Then by definition \(v(s) = v(\mathcal{I})-v(a)\), hence it is linear in some neighborhood of \(v\in \mathrm{QM}(Y,E)\) if \((Y,E)\) is adapted to \(v\). Moreover, \(v\mapsto v(s)\) is linear on any log smooth model \(\pi\colon(Y,E)\to X\) adapted to \(v\) such that \(E+\pi^{-1}(Z)+\{\pi^*a=0\}\) is a divisor with SNC support. ◻

Proof. This follows from GKKP-extend-forms? or KS-extend-forms?. ◻

Proof. By definition, \[A_X(D)=\mathrm{ord}_D(\pi^*\sigma^{n/p})\ge \frac{n}{p}\cdot \mathrm{ord}_D(\pi^*\sigma) = \frac{n}{p}\cdot A_\sigma (D) = \frac{nr}{p}\, ,\] with equality if and only if \(\tilde{\sigma}^{n/p} \in H^0(Y,\Omega^{[n]}_{Y}(\log D)(-\frac{nr}{p}D))\) does not vanish at the generic point of \(D\). This proves that \((2)\Leftrightarrow (3)\).

To prove \((1)\Leftrightarrow (3)\), we track how the differential form specializes using Lemma 14. We follow the notation thereof: let \(\mathscr{Y}=Y\times \mathbb{A}^1\), \(\varphi\colon \mathscr{Y}'\to \mathscr{Y}\) be the normalized blowup along \(D\times \{0\}\subseteq \mathscr{Y}\), and \(\mathscr{G}\) be the exceptional divisor over the generic point \(\eta\) of \(D\times \{0\}\). Let \(\mathscr{D}=D\times \mathbb{A}^1\subseteq \mathscr{Y}\). Then \(\mathrm{pr}_Y^*\tilde{\sigma}\) is a section of \[\mathrm{pr}_Y^*\Omega^{[p]}_{Y}(\log D)(-rD) = \Omega^{[p]}_{\mathscr{Y}/\mathbb{A}^1}(\log \mathscr{D})(-r\mathscr{D})\] that does not vanish at \(\eta\), thus \[\sigma_{\mathscr{Y}'} = \varphi^*\mathrm{pr}_Y^*\tilde{\sigma} \in \varphi^*\Omega^{[p]}_{\mathscr{Y}/\mathbb{A}^1}(\log \mathscr{D})(-r\mathscr{D})\] does not vanish at the generic point \(\zeta\) of \(\mathscr{G}\) by construction. By Proposition 29(2), in order to check whether the specialization \(\sigma_0\) of \(\sigma\) along the test configuration \(\mathscr{X}\to \mathbb{A}^1\) is non-degenerate, it suffices to show that \(\sigma_{\mathscr{X}}^{n/p}\) is non-vanishing at the generic point of \(\mathscr{X}_0\). By Lemma 14, \(\mathscr{X}_0\) is the strict transform of \(\mathscr{G}\). By a local computation, we have \[\varphi^*\Omega^{[p]}_{\mathscr{Y}/\mathbb{A}^1}(\log \mathscr{D})(-r\mathscr{D}) = \Omega^{[p]}_{\mathscr{Y}'/\mathbb{A}^1}(\log \mathscr{G})(-r\mathscr{G}) \cong \Omega^{[p]}_{\mathscr{Y}'/\mathbb{A}^1}(-r\mathscr{G})\] in a neighborhood of \(\zeta\), where the second isomorphism holds because \(\mathscr{G}=\{t=0\}\) at \(\zeta\) (as before \(t\) is the coordinate on \(\mathbb{A}^1\)). Thus by definition, the integer \(m\) in Definition 27 is equal to \(-r\), and \(\sigma_{\mathscr{X}} = \sigma_{\mathscr{Y}'}\) at the generic point \(\zeta\) of \(\mathscr{G}\). It now suffices to check that \(\sigma_{\mathscr{Y}'}^{n/p}\) is non-vanishing at \(\zeta\). But \[\sigma_{\mathscr{Y}'}^{n/p} = \varphi^*\mathrm{pr}_Y^*(\tilde{\sigma}^{n/p}),\] and the right hand side does not vanish at \(\zeta\) if and only if \(\mathrm{pr}_Y^*(\tilde{\sigma}^{n/p})\) does not vanish at \(\eta = \varphi(\zeta)\). This is the case if and only if \(\tilde{\sigma}^{n/p}\) does not vanish at the generic point of \(D\). This proves that \((1)\Leftrightarrow (3)\). ◻

Proof. By Lemma 33, it suffices to prove this for quasi-regular Reeb vectors, thus we may assume that \(\mathbb{T}=\mathbb{G}_m\), the \(\mathbb{G}_m\)-action on \(\sigma\) has weight \(\ell\in\mathbb{Z}\), and we need to show that \(A_\sigma (D) = \ell\), where \(D\) is the prime divisor over \(X\) corresponding to this \(\mathbb{G}_m\)-action. Recall that \(g\colon X\setminus \{x\}\to Z:=(X\setminus \{x\})/\mathbb{G}_m\) has a Seifert \(\mathbb{G}_m\)-bundle structure and \(D\) can be identified with its zero section. Let \(Y\) be the variety obtained by adding the zero section to \(X\setminus \{x\}\); in particular \(X\setminus\{x\} \cong Y\setminus D\) and we get a morphism \(Y\to X\) that contracts \(D\) to \(x\). The \(\mathbb{G}_m\)-invariant differential forms on \(Y\setminus D\) are generated by linear combinations of wedge products of \(\frac{\mathrm{d}t}{t}\) and \(g^*\sigma_0\), where \(t\) is a \(\mathbb{G}_m\)-equivariant defining equation of \(D\) and \(\sigma_0\) varies over the \(1\)-forms on \(Z\). These are also the local free generators of \(\Omega_Y^{[p]}(\log D)\) at the generic point of \(D\). Since \(t\) has weight \(1\), we see that \(t^{-\ell}\sigma\) is \(\mathbb{G}_m\)-invariant, thus by the above discussion it defines a local section of \(\Omega_Y^{[p]}(\log D)\) that does not vanish at the generic point of \(D\). It follows that \(A_\sigma(D)=\ell\) by the definition of log discrepancy.

For the last statement, note that if \(\sigma\) is a symplectic form, then \(\sigma^{n/2}\) is a free generator of \(\omega_X\), hence \(A_{\sigma^{n/2}}(D)=A_X(D)>0\) by the definition of log discrepancy and the fact that \(X\) is klt. It follows that the \(\mathbb{G}_m\)-action has positive weight on \(\sigma^{n/2}\) and hence on \(\sigma\) as well. ◻

Proof. To make sense of the statement, we first observe that for any vector field \(\eta\) on \(\mathbb{A}^1\) and any lift \(\xi\) of \(\eta\) to some open subset \(\mathscr{U}\) of \(\mathscr{X}\), we have \(\mathcal{L}_{\xi}(f^*\mathrm{d}t) = f^*\mathcal{L}_\eta (\mathrm{d}t)\in f^*\Omega_{\mathbb{A}^1}^1\), thus the Lie derivative \(\mathcal{L}_\xi\) on \(\Omega_{\mathscr{U}/\mathbb{A}^1}^{[\bullet]}\) is well defined.

By construction, the symplectic form \(\sigma_0\) is \(\mathbb{G}_m\)-equivariant. By Lemma 38, the \(\mathbb{G}_m\)-action has positive weight on \(\sigma_0\), hence also on \(\sigma_{\mathscr{X}}\). From 14 , we also see that the \(\mathbb{G}_m\)-action on \(t\) (resp. \(\partial_t\)) has weight \(-1\) (resp. \(1\)).

Let \(f\colon \mathscr{Y}\to \mathscr{X}\) be a terminal modification Kol13?. Then \(A_{\mathscr{X}}(\mathscr{E})=1\) for every exceptional divisor \(\mathscr{E}\) of \(f\). Since \(\mathscr{X}\) has a klt fiber over \(0\in \mathbb{A}^1\), the pair \((\mathscr{X},\mathscr{X}_0)\) is plt by inversion of adjunction KM98?, thus \(A_{\mathscr{X}}(\mathscr{E}) = A_{\mathscr{X},\mathscr{X}_0}(\mathscr{E})+\mathrm{ord}_{\mathscr{E}}(\mathscr{X}_0)>\mathrm{ord}_{\mathscr{E}}(\mathscr{X}_0)\). It follows that \(\mathrm{ord}_{\mathscr{E}}(\mathscr{X}_0) = 0\) and hence every exceptional divisor of \(f\) dominates \(\mathbb{A}^1\). Moreover, we get \(f^*(K_{\mathscr{X}}+\mathscr{X}_0)=K_{\mathscr{Y}}+\mathscr{Y}_0\), which implies that \((\mathscr{Y},\mathscr{Y}_0)\) is also plt, hence \(\mathscr{Y}_0\) is normal by KM98?.

Let \(\mathscr{U}\subseteq \mathscr{Y}\) be the (\(\mathbb{G}_m\)-invariant) locus where the induced morphism \(\mathscr{Y}\to \mathbb{A}^1\) is smooth. Then its complement \(\mathscr{Y}\setminus \mathscr{U}\) has codimension at least \(3\) because \(\mathscr{Y}\) is terminal and \(\mathscr{Y}_0\) is normal. Because \(f\colon \mathscr{Y}\to \mathscr{X}\) is crepant and fiberwise birational, the pullback \(\sigma_{\mathscr{Y}}\) of \(\sigma_{\mathscr{X}}\) remains symplectic on each fiber \(\mathscr{Y}_t\). Since both \(\mathscr{X}\) and \(\mathscr{Y}\) have rational singularities (see KM98?) and \(\mathscr{X}\) is affine, we have \(H^i(\mathscr{Y},\mathcal{O}_{\mathscr{Y}})=H^i(\mathscr{X},\mathcal{O}_{\mathscr{X}})=0\) for \(i=1,2\). Since \(\mathscr{Y}\) is Cohen-Macaulay by KM98?, from the long exact sequence of local cohomology we also have \(H^i(\mathscr{U},\mathcal{O}_{\mathscr{U}})=0\) for \(i=1,2\).

Consider the truncated algebraic de Rham complex \(\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1}\) on \(\mathscr{U}\) (by convention the complex starts in degree \(1\)): \[\Omega_{\mathscr{U}/\mathbb{A}^1}^1 \xrightarrow{\mathrm{d}} \Omega_{\mathscr{U}/\mathbb{A}^1}^2 \xrightarrow{\mathrm{d}} \dots \xrightarrow{\mathrm{d}} \Omega_{\mathscr{U}/\mathbb{A}^1}^n .\] It is well known that this complex controls the deformation theory of smooth symplectic varieties. The \(\mathbb{G}_m\)-action on \(\mathscr{X}\) lifts to a \(\mathbb{G}_m\)-action on \(\mathscr{U}\), hence \(\mathbb{G}_m\) also naturally acts on \(\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1}\) and its hypercohomology.

We show that the obstruction to the equivariant lifting problem of \(\partial_t\) in the lemma’s statement is a class of positive weight in \(\mathbb{H}^2(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1})\). To see this, let \((\mathscr{U}_i)_{i\in I}\) be a finite open covering of \(\mathscr{U}\) by \(\mathbb{G}_m\)-invariant affine subsets (this is possible by Sumihiro?). Since each term in the de Rham complex is quasi-coherent, the hypercohomology groups \(\mathbb{H}^\bullet(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1})\) can be computed by the total Čech complex \((\mathscr{C}^\bullet,\delta)\) stacks-project? (see Namikawa-deformation-terminal?, whose degree \(m\) term is \[\mathscr{C}^m = \bigoplus_{p+q=m,\,q\ge 1} \prod_{i_0<\dots<i_p} H^0(\mathscr{U}_{i_0\dots i_p},\Omega_{\mathscr{U}/\mathbb{A}^1}^q).\] Since \(\mathscr{U}\to \mathbb{A}^1\) is smooth, on each \(\mathbb{G}_m\)-invariant affine open \(\mathscr{U}_i\) we can lift \(\partial_t\) to some \(\mathbb{G}_m\)-equivariant vector fields \(\xi_i\) of weight \(1\) (i.e. same weight as \(\partial_t\)). On each \(\mathscr{U}_{ij}\) we then get a vertical vector field \(\xi_{ij}=\xi_i - \xi_j \in T_{\mathscr{U}/\mathbb{A}^1}\). Since contraction \(\iota_{\bullet}(\sigma_{\mathscr{U}})\) with the relative symplectic form gives an isomorphism \(T_{\mathscr{U}/\mathbb{A}^1}\cong \Omega^1_{\mathscr{U}/\mathbb{A}^1}\), and since \(\sigma_{\mathscr{U}}\) has positive weight under the \(\mathbb{G}_m\)-action, the collection \[\theta:=\big( \iota_{\xi_{ij}}(\sigma_{\mathscr{U}}), \mathcal{L}_{\xi_i} \sigma_{\mathscr{U}}\big) \in \mathscr{C}^2\] gives a \(\mathbb{G}_m\)-equivariant element with positive weight.

We show that \(\delta(\theta)=0\). This amounts to three equalities: \[\begin{align} \mathrm{d}(\mathcal{L}_{\xi_i} \sigma_{\mathscr{U}}) = \mathcal{L}_{\xi_i} (\mathrm{d}\sigma_{\mathscr{U}}) & = 0, \\ \mathcal{L}_{\xi_{ij}} \sigma_{\mathscr{U}} - \mathrm{d}\, \iota_{\xi_{ij}}(\sigma_{\mathscr{U}}) = \iota_{\xi_{ij}}(\mathrm{d}\sigma_{\mathscr{U}}) & = 0, \\ \iota_{\xi_{ij}}(\sigma_{\mathscr{U}}) + \iota_{\xi_{jk}}(\sigma_{\mathscr{U}}) + \iota_{\xi_{ki}}(\sigma_{\mathscr{U}}) & = 0 \quad (on \mathscr{U}_{ijk}). \end{align}\] The first equality holds because \(\mathrm{d}\sigma_{\mathscr{X}}=0\), the second follows from Cartan’s formula \(\mathcal{L}_\xi = \mathrm{d}\, \iota_\xi+\iota_\xi\, \mathrm{d}\), while the third holds as \(\xi_{ij}+\xi_{jk}+\xi_{ki}=0\) by construction. Thus we get a positive-weighted cohomology class \([\theta]\in H^2(\mathscr{C}^\bullet)\cong \mathbb{H}^2(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1})\). Different choices of lifting have the form \(\xi'_i=\xi_i+\eta_i\) for some \(\eta_i\in H^0(T_{\mathscr{U}_i/\mathbb{A}^1})\), and it is straightforward to check using Cartan’s formula that the corresponding element \(\theta'\in \mathscr{C}^2\) satisfies \(\theta'-\theta = \delta(\tau)\) where \(\tau:=\big(\iota_{\eta_i}(\sigma_{\mathscr{U}})\big)\in \mathscr{C}^1\). From these we conclude that the cohomology class \([\theta]\) is independent of the choice of the lifting \(\xi_i\).

Next we show that \([\theta]=0\) if and only if \(\partial_t\) equivariantly lifts to some globally defined vector field \(\xi\) such that \(\mathcal{L}_{\xi} \sigma_{\mathscr{U}}=0\). Indeed, \([\theta]=0\) implies that we can find lifts \(\xi_i\) such that \(\iota_{\xi_{ij}}(\sigma_{\mathscr{U}}) =0\) and \(\mathcal{L}_{\xi_i} \sigma_{\mathscr{U}}=0\). Since contraction \(\iota_{\bullet}(\sigma_{\mathscr{U}})\) with the relative symplectic form gives an isomorphism \(T_{\mathscr{U}/\mathbb{A}^1}\cong \Omega^1_{\mathscr{U}/\mathbb{A}^1}\), this implies \(\xi_{ij}=0\). Therefore, the \(\xi_i\)’s glue to a vector field \(\xi\) and we have \(\mathcal{L}_{\xi} \sigma_{\mathscr{U}}=0\). The reverse direction is clear. It remains to show that \([\theta]=0\). In fact, we shall prove the stronger statement that the weight-positive part of \(\mathbb{H}^2(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1})\) is trivial, cf. Namikawa-deformation-terminal?. For this we follow the argument in ibid. Since \(H^i(\mathscr{U},\mathcal{O}_{\mathscr{U}})=0\) for \(i=1,2\), the long exact sequence associated to the distinguished triangle \(\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1} \to \Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet}\to \mathcal{O}_{\mathscr{U}} \to \Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1}[1]\) gives \[\mathbb{H}^2(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1}) \cong \mathbb{H}^2(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet}).\] But the de Rham complex \(\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet}\) is \(\mathbb{G}_m\)-equivariantly quasi-isomorphic to \(f^{-1}\mathcal{O}_{\mathbb{A}^1}\), and the weight positive part of the latter is trivial (because the \(\mathbb{G}_m\)-action on \(\mathbb{A}^1\) has negative weight). Therefore, the weight-positive part of \(\mathbb{H}^2(\mathscr{U},\Omega_{\mathscr{U}/\mathbb{A}^1}^{\bullet\ge 1})\) is also trivial. This concludes the proof. ◻

Proof of Theorem 37. By Lemma 39, we can fix a \(\mathbb{G}_m\)-equivariant lifting \(\xi\) of \(\partial_t\) with \(\mathcal{L}_\xi (\sigma_{\mathscr{X}})=0\). This corresponds to a derivation \(\partial_\xi\colon \mathcal{R}\to \mathcal{R}\) such that \(\partial_\xi(\mathcal{R}_m)\subseteq \mathcal{R}_{m+1}\) for all \(m\in\mathbb{Z}\) and \(\partial_\xi(t) = 1\). The \(\mathbb{G}_m\)-action on \(\mathscr{X}\) also generates a vector field \(\xi'\) whose image in \(\mathbb{A}^1\) is \(-t\frac{\partial}{\partial t}\); as a derivation, it acts by \(\partial_{\xi'}=m\cdot \mathrm{id}\) on \(\mathcal{R}_m\). Then \(\eta = t\xi+\xi'\) satisfies \[\partial_\eta(t)= t\cdot \partial_\xi(t)-t=0 \, ,\] hence it is a \(\mathbb{G}_m\)-invariant vertical vector field (i.e. \(\eta\in H^0(T_{\mathscr{X}/\mathbb{A}^1})\)). In particular, by restricting to the general fiber it defines a derivation \(\partial_\eta\) of \(R\), which can be identified with the derivation \(t\partial_\xi\) on \(\mathcal{R}_0 = R\). The condition \(\partial_\xi(\mathcal{R}_m)\subseteq \mathcal{R}_{m+1}\) is then equivalent to \((\partial_\eta - m)\mathfrak{a}_m\subseteq \mathfrak{a}_{m+1}\). In particular, \(\partial_\eta(\mathfrak{a}_m)\subseteq \mathfrak{a}_m\). Note that every \(\mathfrak{a}_m\) is \(\mathfrak{m}_x\)-primary and \(\mathfrak{a}_1=\mathfrak{m}_x\) since the Kollár component \(D\) is centered at \(x\). Hence the vector field \(\eta\) fixes the canonical section \(\overline{\{x\}\times (\mathbb{A}^1\setminus\{0\})}\) of the test configuration and \(\partial_\eta\) extends to a derivation of the formal completion \(\widehat{R}\). By induction, we see that \[\label{eq:derivation} \frac{(\partial_\eta-m)\cdots(\partial_\eta-m-i+1)}{i!}\cdot \mathfrak{a}_m\subseteq \mathfrak{a}_{m+i}\tag{15}\] for all \(i\in\mathbb{N}\) (by convention, the operator on the left hand side is the identity when \(i=0\)). Let \[\widehat{R}_m:=\{s\in \widehat{R}\,|\,\partial_\eta(s) = ms\}.\] Then \(\widehat{R}_m\cdot \widehat{R}_\ell \subseteq \widehat{R}_{m+\ell}\), as for any \(s\in \widehat{R}_m\) and \(s'\in \widehat{R}_\ell\), we have \[\partial_\eta(s\cdot s')=\partial_\eta s \cdot s'+s\cdot \partial_\eta s' =ms\cdot s'+\ell s\cdot s'=(m+\ell) s\cdot s'\, .\]

We claim that \(\widehat{R}_m\subseteq \widehat{\mathfrak{a}}_m:=\mathfrak{a}_m \widehat{R}\) and the induced map \[\varphi_m\colon \widehat{R}_m\to \widehat{\mathfrak{a}}_m/\widehat{\mathfrak{a}}_{m+1}\cong \mathfrak{a}_m/\mathfrak{a}_{m+1} = \mathrm{gr}^m_D R\] is an isomorphism. Essentially this holds because \(\partial_\eta-m\) is the zero map on \(\mathrm{gr}_D^m R\), and is invertible on \(\mathrm{gr}^{\ell}_D R\) whenever \(\ell\neq m\). More concretely, for any \(s\in \mathfrak{a}_m\), let \(\bar{s}\in \mathfrak{a}_m/\mathfrak{a}_{m+1}\) be its reduction, and set \[\tilde{s}:=\sum_{i=0}^\infty \frac{(-1)^i}{i!} (\partial_\eta-m)\cdots(\partial_\eta-m-i+1)\cdot s \in \widehat{\mathfrak{a}}_m \, ,\] which is well-defined by 15 . Then \(\tilde{s}-s\in \widehat{\mathfrak{a}}_{m+1}\) and \[\begin{align} (\partial_\eta-m)(\tilde{s}) & = \sum_{i=0}^\infty \frac{(-1)^i}{i!} (\partial_\eta-m)\cdots(\partial_\eta-m-i+1) \Big((\partial_\eta-m-i) +i\Big)\cdot s \\ & = \sum_{i=0}^\infty \frac{(-1)^i}{i!} \prod_{j=0}^i (\partial_\eta-m-j) \cdot s + \sum_{i=0}^\infty \frac{(-1)^{i-1}}{i!} \prod_{j=0}^i (\partial_\eta-m-j)\cdot s = 0. \end{align}\] Therefore, \(\tilde{s} \in \widehat{R}_m\) and \(\varphi_m(\tilde{s})=\bar{s}\in \mathfrak{a}_m/\mathfrak{a}_{m+1}\). This proves that \(\varphi_m\) is surjective. On the other hand, if \(0\neq s\in \widehat{R}_m\) then there is some integer \(\ell\in \mathbb{N}\) (in fact \(\ell = \mathrm{ord}_D(s)\)) such that \(s\in \widehat{\mathfrak{a}}_\ell \setminus \widehat{\mathfrak{a}}_{\ell+1}\). In particular, its reduction \(\bar{s}\in \widehat{\mathfrak{a}}_\ell/\widehat{\mathfrak{a}}_{\ell+1}\cong \mathrm{gr}^\ell_D R\) is nonzero and we have \(\partial_\eta (\bar{s})=\ell \bar{s}\). But we also have \(\partial_\eta(s)=ms\), hence \(\ell = m\). This gives \(\widehat{R}_m\subseteq \widehat{\mathfrak{a}}_m\) as well as the injectivity of \(\varphi_m\). The claim is thus proved.

Using the isomorphism \(\varphi_m\) we may identify \(R_0=\bigoplus_{m\in\mathbb{N}} \mathrm{gr}_D^m R\) with the subring \(\bigoplus_{m\in\mathbb{N}} \widehat{R}_m\) of \(\widehat{R}\) whose completion is also \(\widehat{R}\). Moreover, \(\widehat{\mathfrak{a}}_m = \widehat{\bigoplus}_{\ell\ge m} \widehat{R}_\ell\). As a result, we get a formal \(\mathbb{G}_m\)-action on \(\widehat{R}\) with corresponding Kollár component \(D\) and \(\widehat{R}\cong \widehat{R_0}\). It remains to show that the induced symplectic form \(\widehat{\sigma}\) on \(\widehat{X}_x = \mathrm{Spec}(\widehat{R})\) is equivariant under the above \(\mathbb{G}_m\)-action. Equivalently, we need to show that \(\mathcal{L}_\eta(\sigma_{\mathscr{X}})=\ell \cdot \sigma_\mathscr{X}\) for some integer \(\ell\). This follows because by our choice of the vector fields \(\xi,\xi'\), we have \(\mathcal{L}_{\xi'}(\sigma_{\mathscr{X}})=\ell \cdot \sigma_\mathscr{X}\) where \(\ell = \mathrm{wt}(\sigma_{\mathscr{X}})\), while \(\mathcal{L}_{t\xi}(\sigma_{\mathscr{X}}) = t\mathcal{L}_{\xi}(\sigma_{\mathscr{X}}) = 0 \in H^0(\Omega^{[2]}_{\mathscr{X}/\mathbb{A}^1})\) by Cartan’s formula. This completes the proof. ◻

Proof. This should be well-known to experts. The exact sequence \[0\to \Omega^1_{\mathcal{X}}\to \Omega^1_{\mathcal{X}}(\log \mathcal{D})\xrightarrow{\mathrm{res}}\mathcal{O}_{\mathcal{D}}\to 0\] on \(\mathcal{X}\) restricts to an exact sequence \[\Omega_{\mathcal{X}}^1|_{\mathcal{D}}\to \Omega^1_{\mathcal{X}}(\log \mathcal{D})|_{\mathcal{D}}\to \mathcal{O}_{\mathcal{D}}\to 0\] on \(\mathcal{D}\). The first map factors through \(\Omega_{\mathcal{D}}^1\), and we get an induced exact sequence (exactness can be checked in local coordinates) \[0\to \Omega_{\mathcal{D}}^1\to \Omega^1_{\mathcal{X}}(\log \mathcal{D})|_{\mathcal{D}}\to \mathcal{O}_{\mathcal{D}}\to 0.\] A direct computation using Čech cohomology then shows that the corresponding extension class is \(c_1(N_{{\mathcal{D}}/\mathcal{X}})\in H^1(\mathcal{D},\Omega_{\mathcal{D}}^1)\). (Here we note that as we work over \(\mathbb{C}\), any DM stack is tame, see e.g. AOV-tame-stack?, thus the cohomology of a quasi-coherent sheaf can be computed on the coarse moduli space and therefore coincides with Čech cohomology.) ◻

Proof. By construction, \(\mathcal{D}\) is a Cartier divisor on \(\mathcal{X}\). Since \(f\) is quasi-étale, we have \(f^*(K_X+D)=K_{\mathcal{X}}+\mathcal{D}\). As \((X,D)\) is plt, we deduce that \((\mathcal{X},\mathcal{D})\) is also plt KM98?, hence (1) holds by KM98?. By adjunction we get \(K_{\mathcal{D}} = g^*(K_D+\Delta_D)\), thus \(g^*\Delta_D\) is an integral divisor on \(\mathcal{D}\) which gives (2).

By definition, both \(\Omega^{[1]}_{(D,\Delta_D,g)}\) and \(\Omega^{[1]}_{\mathcal{D}}\) are reflexive subsheaves of \(\Omega^{[1]}_{\mathcal{D}}(f^*\Delta_D)\). To prove the first isomorphism in (3), it suffices to check that they agree over every codimension one point (call it \(\xi\)) of \(D\). Since \((X,D)\) is plt, étale locally at \(\xi\) it is a cyclic quotient \(\big((\mathbb{A}^2,\{x=0\})/\mu_r\big)\times \mathbb{A}^{n-2}\) with \(n=\dim X\) (see e.g. Kol13?), thus (ignoring the smooth factor \(\mathbb{A}^{n-2}\)) we reduce to the case \(\mathcal{D}=[\mathbb{A}^1/\mu_r]\), with coarse moduli space \(D=\mathbb{A}^1/\mu_r\cong \mathbb{A}^1\), and \(\Delta_D = (1-\frac{1}{r})[0]\) (see Kol13?), where the equality \(\Omega^{[1]}_{(D,\Delta_D,g)}=\Omega^{[1]}_{\mathcal{D}}\) holds by a direct local computation. This gives the first part of (3).

By definition, \(E_{(D,\Delta_D,g),N_{D/X}}\) is the extension of \(\mathcal{O}_{\mathcal{D}}\) by \(\Omega^{[1]}_{(D,\Delta_D,g)}\) with extension class \(g^*c_1(N_{D/X})\). As \(\Omega^{[1]}_{(D,\Delta_D,g)} \cong \Omega^{[1]}_{\mathcal{D}}\) and \(f^*D = \mathcal{D}\), we see that \(E_{(D,\Delta_D,g),N_{D/X}}\) is also the extension of \(\mathcal{O}_{\mathcal{D}}\) by \(\Omega^{[1]}_{\mathcal{D}}\) with extension class \(c_1(N_{\mathcal{D}/\mathcal{X}})\), thus the second part of (3) holds as well. ◻

Proof. See Theorem 72. ◻

Proof. The equality \(\mu_H(E_{X,\Delta})=-\frac{1}{n+1}H^n\) is a direct consequence of CKT-hyperbolicity?. If \((X,\Delta)\) is K-semistable, then \(E_{X,\Delta}\) is slope semistable with respect to \(H\) by Theorem 47, thus \[\mu_{\max,H}(E_{X,\Delta}) = \mu_{H}(E_{X,\Delta}) = -\frac{1}{n+1}(H^n)\] and ?? obviously holds.

Assume that \((X,\Delta)\) is not K-semistable and let \(\delta=\delta(X,\Delta)\). Then \(0<\delta<1\) and \(\delta\in \mathbb{Q}\) by LXZ-HRFG?. By LXZ-HRFG?, there exists some general \(\mathbb{Q}\)-divisor \(0\le G\sim_{\mathbb{Q}} H\) such that the log Fano pair \((X,\Gamma:=\Delta+(1-\delta)G)\) is K-semistable.

Let \(f\colon Y\to X\) be a finite cover adapted to \((X,\Gamma)\). Since \(G\) is general, \(f\) is also adapted to \((X,\Delta)\). By construction, \(\Omega^{[1]}_{(X,\Delta,f)}\) is a subsheaf of \(\Omega^{[1]}_{(X,\Gamma,f)}\). Since \(K_X+\Gamma\sim_{\mathbb{Q}} -\delta H\) is proportional to \(K_X+\Delta\), we deduce that the canonical extension \(E_{(X,\Gamma,f)}\) can be obtained as the cokernel of \[\Omega^{[1]}_{(X,\Delta,f)} \to E_{(X,\Delta,f)} \oplus \Omega^{[1]}_{(X,\Gamma,f)},\] and we have a commutative diagram \[\begin{tikzcd} 0 \arrow[r] & \Omega^{[1]}_{(X,\Delta,f)} \arrow[r] \arrow[d, hook] & E_{(X,\Delta,f)} \arrow[r] \arrow[d, hook] & \mathcal{O}_Y \arrow[r] \arrow[d, equal] & 0 \\ 0 \arrow[r] & \Omega^{[1]}_{(X,\Gamma,f)} \arrow[r] & E_{(X,\Gamma,f)} \arrow[r] & \mathcal{O}_Y \arrow[r] & 0 \end{tikzcd}\] whose middle column is injective by the five lemma. By Theorem 47, we know that \(E_{X,\Gamma}\) is slope semistable with respect to \(-(K_X+\Gamma)\), hence also slope semistable with respect to \(H\). It follows that \[\mu_{\max,H}(E_{X,\Delta})\le \mu_{\max,H}(E_{X,\Gamma}) = \mu_H (E_{X,\Gamma}) = \frac{1}{n+1}(K_X+\Gamma\cdot H^{n-1}) = -\frac{\delta}{n+1}(H^n).\] This proves ?? . ◻

Proof. We first prove (1). Let \(\delta := \min\{\delta(D,\Delta_D),1\}\) and \(r := A_\sigma(D)\). Let \(Y\to X\) be the plt blowup of \(D\). Consider the index one covering stack \(\varphi\colon \mathcal{Y}\to Y\) with respect to \(D\) and let \(\mathcal{D}=\varphi^*D\). By pulling back through \(\varphi\), we shall view the strict transform \(\tilde{\sigma}\) of \(\sigma\) (Definition 35) as a section of \(\Omega_{\mathcal{Y}}^{[p]}(\log \mathcal{D})(-r\mathcal{D})\) that does not vanish at the generic point of \(\mathcal{D}\). By restriction, we get a non-zero section \(\tilde{\sigma}|_{\mathcal{D}}\) of \(\Omega_{\mathcal{Y}}^{[p]}(\log \mathcal{D})(-r\mathcal{D})|_{\mathcal{D}}\). In particular, \[H^0(\mathcal{D},\Omega_{\mathcal{Y}}^{[p]}(\log \mathcal{D})(-r\mathcal{D})|_{\mathcal{D}}) \neq 0\] and hence \[\mu_{\max}(\Omega^{[p]}_{\mathcal{Y}}(\log \mathcal{D})(-r\mathcal{D})|_{\mathcal{D}}) \ge 0,\] where the slope is computed with respect to \(-(K_D+\Delta_D)\) and defined in a way analogous to Definition 46 by considering finite covers of \(D\) that factor through \(\mathcal{D}\). Since \[K_D+\Delta_D = (K_Y+D)|_D\sim_\mathbb{Q}(\pi^*K_X + A_{X}(D)\cdot D)|_D\sim_\mathbb{Q}A_{X}(D)\cdot D|_D,\] we also have \[\begin{align} \mu_{\max}(\Omega^{[p]}_{\mathcal{Y}}(\log \mathcal{D})(-r\mathcal{D})|_{\mathcal{D}}) & \le p\cdot \mu_{\max}(\Omega^{[1]}_{\mathcal{Y}}(\log \mathcal{D})|_{\mathcal{D}})+ (-rD|_D\cdot (-K_D-\Delta_D)^{n-2}) \\ & = p\cdot \mu_{\max}(\Omega^{[1]}_{\mathcal{Y}}(\log \mathcal{D})|_{\mathcal{D}})+\frac{r}{A_{X}(D)}(-K_D-\Delta_D)^{n-1}\,. \end{align}\] By Lemmas 44 and 45, we have \[\Omega_{\mathcal{Y}}^{[1]}(\log \mathcal{D})|_{\mathcal{D}} \cong E_{\mathcal{D},N_{\mathcal{D}/\mathcal{Y}}}= E_{(D,\Delta_D,\varphi|_{\mathcal{D}}),N_{D/Y}}\] (since the sheaves in question are reflexive, it suffices to show that they agree on the big open set where both \(\mathcal{Y}\) and \(\mathcal{D}\) are smooth, thus Lemmas 44 applies). Since \(K_D+\Delta_D\) is proportional to \(D|_D\), we see that \(E_{(D,\Delta_D,\varphi|_{\mathcal{D}}),N_{D/Y}}\) is isomorphic to the canonical extension \(E_{D,\Delta_D,\varphi}\), thus \[\mu_{\max}(\Omega^{[1]}_{\mathcal{Y}}(\log \mathcal{D})|_{\mathcal{D}})=\mu_{\max} (E_{D,\Delta_D}) \, .\] Therefore, combining the inequalities above we conclude \[A_{\sigma}(D)= r\ge \frac{-p\cdot \mu_{\max}(E_{D,\Delta_D}) \cdot A_{X}(D)}{(-K_D-\Delta_D)^{n-1}} \, .\] By Lemma 48, we also have \[\mu_{\max}(E_{D,\Delta_D})\le -\frac{\delta}{n}(-K_D-\Delta_D)^{n-1},\] hence \(A_{\sigma}(D)\ge \frac{\delta \cdot p}{n} A_X(D)\) which gives (1).

For (2), let \((Y,E)\to X\) be a log smooth model adapted to \(v\). By Theorem 22, for any \(\varepsilon > 0\) there exists a divisorial valuation \(\lambda \cdot \mathrm{ord}_D \in \mathrm{QM}(Y,E)\) near \(v\) such that \(D\) is a Kollár component and \(\delta(D,\Delta_D) \ge 1 - \varepsilon\). By part (1), \[(1-\varepsilon) \cdot \frac{A_X(\lambda\cdot \mathrm{ord}_D)}{n} \le \frac{A_\sigma(\lambda\cdot \mathrm{ord}_D)}{p}\, .\] By Lemma 33, we may assume that both \(A_X(\cdot)\) and \(A_\sigma(\cdot)\) are linear around \(v\in \mathrm{QM}(Y,E)\). Letting \(\varepsilon \to 0\), we obtain \(\frac{A_X(v)}{n}\le \frac{A_{\sigma}(v)}{p}\). ◻

Proof. Let \(n=\dim X\). We first show that \[\label{eq:n47p32log32discrep} \frac{A_X(v)}{n}= \frac{A_{\sigma}(v)}{p}\, .\tag{17}\] Indeed, by Theorem 49(2), we have \(\frac{A_X(v)}{n}\le \frac{A_{\sigma}(v)}{p}\). On the other hand, since \(\sigma\) is non-degenerate, \(\sigma^{\frac{n}{p}}\) is a free generator of \(\omega_X\) (see Remark 26), hence by definition \(A_X(w) = A_{\sigma^{n/p}}(w) \ge \frac{n}{p} A_\sigma(w)\) for any quasi-monomial valuation \(w\in \mathrm{Val}_{X}\). In particular, \(A_X(v) \ge \frac{n}{p} A_\sigma(v)\). Combining the two inequalities we obtain 17 .

By Lemma 33, both \(A_\sigma(\cdot)\) and \(A_X(\cdot)\) are linear in some neighborhood of \(v\) in \(\mathrm{QM}(Y,E)\). Since \(\frac{A_X(w)}{n}\ge \frac{A_{\sigma}(w)}{p}\) for any quasi-monomial valuation \(w\in \mathrm{Val}_X\) and equality holds for \(v\), we must have \[\label{eq:n47p32log32discrep32in32nbhd} \frac{A_X(w)}{n} = \frac{A_{\sigma}(w)}{p}\tag{18}\] in some neighborhood \(U\subseteq \mathrm{QM}(Y,E)\) of \(v\). Shrinking \(U\) if necessary, we may assume by Theorem 22 that every divisorial valuation in \(U\) corresponds to a Kollár component. By LX-higher-rank?, we may also assume (possibly after shrinking \(U\)) that the corresponding test configuration degenerates \(x\in X\) to \(x_0\in X_0\). By 18 and Proposition 36, these Kollár components are also \(\sigma\)-admissible; in other words, the specialization \(\sigma_0\) of \(\sigma\) is also non-degenerate. ◻

Proof. This should be well-known to the experts, see e.g. Kol-cone-auto?. We sketch a proof for the reader’s convenience. Let \(\widehat{R}=\widehat{\mathcal{O}_{X,x}}\cong \widehat{\mathcal{O}_{Y,y}}\). By Kol-cone-auto?, the maximal torus in the formal automorphism group \(\mathrm{Aut}(\widehat{R})\) is well-defined and unique up to conjugation.

If \(\mathbb{T}_X\) is the torus acting on \(x\in X\), we may extend it to a maximal torus \(\mathbb{T}\subseteq \mathrm{Aut}(\widehat{R})\). The subalgebra \(R_X:=H^0(X,\mathcal{O}_X)\subseteq \widehat{R}\) is a direct sum of \(\mathbb{T}_X\)-invariant subspaces. Because the \(\mathbb{T}_X\)-action is good, every \(\mathbb{T}_X\)-invariant subspace of a given weight is finite dimensional, and further decomposes into a direct sum of \(\mathbb{T}\)-invariant subspaces. It follows that \(R_X\) can be recovered from \(\widehat{R}\) as the direct sum of \(\mathbb{T}\)-invariant subspaces. Since \(\mathbb{T}\) is unique up to conjugation, this implies that the algebra \(R_X\) is determined (up to isomorphism) by \(\widehat{R}\), i.e. we have \[\begin{tikzcd} (R_X, \mathbb{T}_X\subset \mathbb{T}) \arrow[r, hook] \arrow[d,dotted] & (\widehat{R},\mathbb{T})\arrow{d}{\cong} \\ (R_Y, \mathbb{T}_Y\subset \mathbb{T}) \arrow[r,hook] & (\widehat{R},\mathbb{T}) \end{tikzcd}\] In particular, there exists a \(\mathbb{T}\)-equivariant algebraic isomorphism between \(X\) and \(Y\). ◻

Proof. For any finitely generated \(\widehat{R}\)-modules \(M,N\) and any \(l\in \mathbb{N}\), let \(V_l\) be the image of \[\varphi_l\colon \mathrm{Hom}(M,N)\to \mathrm{Hom}(M_l,N_l)\] (where for ease of notation we denote \(M\otimes R_l\) by \(M_l\), etc., and Hom is taken in the category of \(\widehat{R}\)-modules). Since \(\mathrm{Hom}(M_l,N_l)\) is of finite dimension and \(\mathrm{Hom}(M,N)\) is an inverse limit of \(\mathrm{Hom}(M_l,N_l)\), we know that \(V_l\) is also the image of \(\mathrm{Hom}(M_{l'},N_{l'})\) for some sufficiently large \(l'\). We claim that for any \(k\in\mathbb{N}\), there exists some \(l\in\mathbb{N}\), \(l\ge k\) such that the induced map \[\varphi_{k,l}\colon\mathrm{Hom}(M,N)\otimes R_k \to V_l\otimes R_k\] is an isomorphism. Indeed, let \(L\to M\) be a surjection from a finite free \(\widehat{R}\)-module, then \(\mathrm{Hom}(M,N)\) is a submodule of \(\mathrm{Hom}(L,N)\) and \(s\in \mathrm{Hom}(M,N)\) is in the kernel of \(\varphi_{l}\) if and only if \(s\in \mathfrak{m}^l \cdot\mathrm{Hom}(L,N)\). By the Artin-Rees lemma, the latter condition implies \(s\in \mathfrak{m}^k\cdot\mathrm{Hom}(M,N)\) if \(l\gg 0\). This translates to the injectivity of \(\varphi_{k,l}\), whereas its surjectivity is clear. It then follows that \(\mathrm{Hom}(M,N)\otimes R_k\) is determined by \(M_l\) and \(N_l\) for some sufficiently large \(l\in\mathbb{N}\) (depending on \(k\)).

By definition, \[\widehat{\Omega}_{X}^{[p]} = \mathrm{Hom}(\mathrm{Hom}(\widehat{\Omega}_{X}^p,\widehat{R}),\widehat{R})\,.\] Applying the above observation twice, we see that \(\Omega_k\) only depends on \(\Omega_{X}^p\otimes R_l\) and \(R_l\) for some sufficiently large \(l\). Note that \(\Omega_{X}^1\otimes R_l\cong \Omega_{R_{l+1}/\mathbb{C}}\otimes R_l\) only depends on \(R_{l+1}\). Therefore, \(\Omega_k\) only depends on \(R_l\) for sufficiently large \(l\). ◻

Proof. By Lemma 56, we can choose an increasing sequence of positive integers \(l_k\ge k\) (\(k=1,2,\dots\)) such that \(\Omega_k\) only depends on \(R_{l_k}\) for each \(k\). Then we can define \[\mathrm{Aut}(R_{l_k},[\sigma_k]) := \{\varphi\in \mathrm{Aut}(R_{l_k})\,|\,\varphi^*\sigma_k = \lambda \sigma_kfor some\lambda\in \mathbb{C}^*\},\] which is an algebraic subgroup of \(\mathrm{Aut}(R_{l_k})\). Recall that \(\Omega_k\) and \(R_k\) only depend on the completion \(\widehat{R}\), and we have \(\mathrm{Aut}(\widehat{R},[\widehat{\sigma}]) = \varprojlim \mathrm{Aut}(R_{l_k},[\sigma_k])\). The kernel of the natural homomorphism \[\mathrm{Aut}(R_{l_{k+1}},[\sigma_{k+1}])\to \mathrm{Aut}(R_{l_k},[\sigma_k])\] is unipotent for all \(k\ge 2\), as it is contained in the kernel of \(\mathrm{Aut}(R_{l_{k+1}})\to \mathrm{Aut}(R_{l_k})\), and the latter is unipotent as in Kol-cone-auto?. By the same argument as in loc. cit., we deduce that a maximal torus of \(\mathrm{Aut}(\widehat{R},[\widehat{\sigma}])\) exists and is unique up to conjugation. ◻

Proof of Proposition 55. We apply the same argument as in the proof of Lemma 54. Extend \(\mathbb{T}_X\) to a maximal torus \(\mathbb{T}\) of \(\mathrm{Aut}(\widehat{R},[\widehat{\sigma}])\). As in the proof of Lemma 54, we know that \(R_X\) is the direct sum of \(\mathbb{T}\)-invariant subspaces of \(\widehat{R}\). Once the inclusion \(R_X\subseteq \widehat{R}\) and hence \(\Omega_X^{[p]}\subseteq \widehat{\Omega}_{\widehat{R}}^{[p]}\) is fixed, the reflexive \(p\)-form \(\sigma_X\) is also uniquely determined by its image \(\widehat{\sigma}\) in \(\widehat{\Omega}_{\widehat{R}}^{[p]}\). Since the maximal torus of \(\mathrm{Aut}(\widehat{R},[\widehat{\sigma}])\) is unique up to conjugation by Lemma 57, the rest of the proof works as in Lemma 54. ◻

Proof of Theorem 52. By Theorem 50 and Proposition 29(1), we can find some Kollár component over \(x\in X\) whose corresponding test configuration degenerates \((x\in X,\sigma)\) to some conical symplectic singularity \((x_0\in X_0,\sigma_0)\) such that \(x_0\in X_0\) is a K-semistable Fano cone. By Theorem 37, there is a formal isomorphism \((x\in X,\sigma)^{\wedge}\cong (x_0\in X_0,\sigma_0)^{\wedge}\).

We next show that the Reeb vector \(\xi_0\) on \(X_0\) corresponding to the normalized volume minimizer can be chosen so that the action of \(\mathbb{T}_0=\langle\xi_0\rangle\) is \(\sigma_0\)-dilating. To this end, let \(\mathbb{T}\) be a maximal torus of \(\mathrm{Aut}(x_0\in X_0,[\sigma_0])\) that contains the \(\sigma_0\)-dilating \(\mathbb{G}_m\) given by its conical structure. In particular, the \(\mathbb{T}\)-action on \(x_0\in X_0\) is good. We aim to show that we can choose \(\mathbb{T}\supseteq \mathbb{T}_0\). If the minimizer \(v_0\in \mathrm{Val}_{X_0,x_0}\) is not of the form \(\mathrm{wt}_\xi\) for any Reeb vector \(\xi\in N(\mathbb{T})_\mathbb{R}\), then there exists a \(\mathbb{T}\)-equivariant log smooth model \((Y_0,E_0)\to X_0\) adapted to \(v_0\) and an open neighborhood \(U\subseteq \mathrm{QM}(Y_0,E_0)\) of \(v_0\) such that none of the valuations in \(U\) is of the form \(\mathrm{wt}_\xi\). By Theorem 50 and Proposition 29(1), we can find some \(\mathbb{T}\)-invariant Kollár component over \(x_0\in X_0\), given by a divisorial valuation in \(U\) sufficiently close to \(v_0\), whose corresponding test configuration induces a trivial degeneration of \(x_0\in X_0\) (since it is already a K-semistable Fano cone) and specializes \(\sigma_0\) to some other symplectic form \(\sigma'_0\) on \(X_0\). Since the test configuration is \(\mathbb{T}\)-equivariant by construction and the Kollár component is not given by any one parameter subgroup of \(\mathbb{T}\) (by our choice of \(U\)), we get an effective dilating symplectic good \(\mathbb{T}\times \mathbb{G}_m\)-action on \((x_0\in X_0,\sigma'_0)\). By Theorem 37, we have a formal isomorphism \[(x_0\in X_0,\sigma_0)^\wedge \cong (x_0\in X_0,\sigma'_0)^\wedge;\] since both symplectic singularities are conical, we may upgrade it into an algebraic isomorphism \[(x_0\in X_0,\sigma_0) \cong (x_0\in X_0,\sigma'_0)\] by Proposition 55. But then we also get an effective dilating symplectic \(\mathbb{T}\times \mathbb{G}_m\)-action on \((x_0\in X_0,\sigma_0)\), contradicting our assumption that \(\mathbb{T}\) is a maximal torus of \(\mathrm{Aut}(x_0\in X_0,[\sigma_0])\). Therefore, \(v_0=\mathrm{wt}_{\xi_0}\) for some \(\xi_0\in N(\mathbb{T})_\mathbb{R}\) and in particular the action of \(\mathbb{T}_0=\langle\xi_0\rangle\subseteq \mathbb{T}\) is dilating since \(\mathbb{T}\) is so.

The remaining part of the theorem follows directly from Proposition 55. ◻

Proof. By Kol-cone-auto?, the natural map \(G\to \mathrm{Aut}(R/\mathfrak{m}_x^2)\) (where the right hand side is the automorphism group of the finite dimensional algebra \(R/\mathfrak{m}_x^2\)) is an embedding, since it has a unipotent kernel and \(G\) is reductive. We may thus view \(G\) and \(\mathbb{T}\) as subgroups of \({\rm GL}(V)\), where \(V=R/\mathfrak{m}_x^2\). We have a weight decomposition \(V = \bigoplus_{\alpha\in M(\mathbb{T})} V_\alpha\) and the Reeb vector \(\xi\) induces a filtration of \(V\) by \[\mathcal{F}_\xi^\lambda V := \bigoplus_{\alpha\in m(\mathbb{T}),\,\langle\alpha,\xi\rangle\ge \lambda} V_\alpha\] which coincides with the filtration induced by \(\mathrm{wt}_\xi\), i.e. \(\bar{s}\in \mathcal{F}_\xi^\lambda V\) if and only if there exists some \(s\in R\) with reduction \(\bar{s}\) such that \(\mathrm{wt}_\xi (s)\ge \lambda\).

Since \(\mathrm{wt}_\xi\) is \(G\)-invariant by assumption, the filtration \(\mathcal{F}_\xi^\bullet V\) is also \(G\)-invariant, hence \(G\) is contained in the parabolic subgroup \(P(\xi)\subseteq {\rm GL}(V)\) that fixes the filtration \(\mathcal{F}_\xi^\bullet V\). In effect, after a change of basis and with \(m=\dim V\) we may assume that \(\mathbb{T}\) is contained in the diagonal subgroup of \({\rm GL}_m\), the Reeb vector \(\xi\) may then be identified with an element \((\xi_1,\dots,\xi_m)\) of \(\mathbb{R}^m\) which without loss of generality can be arranged so that \(\xi_1\ge \dots\ge\xi_m\). Then \[P(\xi) = \{A=(a_{ij})_{1\le i,j\le m}\,|\,a_{ij}=0whenever\xi_i>\xi_j\}\subseteq {\rm GL}_m\] is the subgroup of block upper triangular matrices. From this explicit description, it is straightforward to see that \(\xi\) is in the Lie algebra of the center of the Levi subgroup \[L(\xi) = \{(a_{ij})\in {\rm GL}_m\,|\,a_{ij}=0whenever\xi_i\neq \xi_j\} \subseteq P(\xi)\] of block diagonal matrices. In particular, the torus \(\mathbb{T}\) generated by \(\xi\) is (by definition) contained in the center of \(L(\xi)\). Since \(G\) is reductive, the induced map \(G\subseteq P(\xi)\to L(\xi)\) is injective (note that since we are in characteristic \(0\), the unipotent kernel of \(P(\xi)\to L(\xi)\) can not have any finite subgroup). Therefore, as \(\mathbb{T}\) commutes with \(L(\xi)\), it also commutes with \(G\). ◻

Proof. Since the \(\mathbb{T}_0\)-action on \(x\in X\) is good, \(\mathrm{Aut}^{\mathbb{T}_0}(x\in (X,\Delta))\) is the direct product of \(\mathbb{T}_0\) and the automorphism group of the log Fano pair \[((X,\Delta)\setminus\{x\})/\mathbb{T}_0\, ,\] in particular it is an algebraic group. Note that \(G\subseteq \mathrm{Aut}^{\mathbb{T}_0}(x\in (X,\Delta))\). Let \(H\) be a maximal reductive subgroup of \(\mathrm{Aut}^{\mathbb{T}_0}(x\in (X,\Delta))\) containing \(G\) (in particular \(H^\circ\) is a Levi subgroup of \(\mathrm{Aut}^{\mathbb{T}_0}(x\in (X,\Delta))\)), and let \(\mathbb{T}\) be a maximal torus of \(H\) containing \(\mathbb{T}_0\). Then \(\mathbb{T}\) is a maximal torus of \(\mathrm{Aut}^{\mathbb{T}_0}(x\in (X,\Delta))\) and hence also of \(\mathrm{Aut}(x\in (X,\Delta))\) as \(\mathrm{Aut}^{\mathbb{T}_0}(x\in (X,\Delta))\) is the centralizer of the torus \(\mathbb{T}_0\).

Since all the maximal tori of \(\mathrm{Aut}(x\in (X,\Delta))\) are conjugate to each other by Lemma 54, we may replace \(\xi\) by a conjugate and assume that \(\langle \xi \rangle\subseteq \mathbb{T}\). In particular, \(\langle \xi \rangle\subseteq H\). Since the normalized volume minimizer \(\mathrm{wt}_\xi\) is unique up to scaling XZ-uniqueness?, it is invariant under the action of \({\rm Aut}(x\in (X,\Delta))\). We can thus apply Lemma [l-center] and conclude that \[\langle \xi \rangle \subseteq Z(H)\subseteq \mathrm{Aut}^H(x\in (X,\Delta))\subseteq \mathrm{Aut}^G(x\in (X,\Delta))\,.\] By assumption, \(\mathbb{T}_0\) is a maximal torus of \(\mathrm{Aut}^G(x\in (X,\Delta))\), hence by Humphreys-alg-gp?, the torus \(\langle \xi \rangle\) is conjugate to a subtorus of \(\mathbb{T}_0\). But \(\mathbb{T}_0\cong \mathbb{G}_m\), thus \(\langle\xi\rangle \cong \mathbb{G}_m\) and it is conjugate to \(\mathbb{T}_0\). ◻

Proof. By Theorem 52, there exists some Reeb vector \(\xi\) on \(X\) such that \(\mathrm{wt}_\xi\) is the normalized volume minimizer and \(\langle\xi\rangle\subseteq \mathrm{Aut}(x\in X,[\sigma])\). The rest of the proof then works as in Lemma 59, replacing \(\mathrm{Aut}(x\in (X,\Delta))\) by \(\mathrm{Aut}(x\in X,[\sigma])\), and the use of Lemma 54 by Theorem 52. ◻

Proof. Let \(G=\prod_{i=1}^d G_i\), \(\mathfrak g=\bigoplus_{i=1}^d \mathfrak g_i\) be the decomposition into simple factors and write \(e=(e_1,\dots,e_d)\). By discarding factors \(G_i\) with \(e_i=0\) (which does not affect the isomorphism class of \((\tilde{o} \in \widetilde{O}_e,\sigma)\)), we may assume that every \(e_i\) is nonzero. We have \(O_e = \prod_{i=1}^d O_{e_i}\), \(\widetilde{O}_e = \prod_{i=1}^d \widetilde{O}_{e_i}\) and \(\sigma = \sum_{i=1}^d \mathrm{pr}_i^*\sigma_i\) where \(\sigma_i\) is the Kirillov-Kostant-Souriau form on the normalized orbit closure \(\widetilde{O}_{e_i}\) of \(e_i\in \mathcal{N}(\mathfrak g_i)\).

Let \(H=\mathrm{Aut}^G(\tilde{o} \in \widetilde{O}_e,[\sigma])^\circ\). As \(H\) commutes with the \(G\)-action, it preserves each \(G\)-orbit and in particular \(H\subseteq \mathrm{Aut}^G(O_e,[\sigma])\). Since \(O_e=G/G_e = \prod_{i=1}^d G_i/G_{i,e_i}\), where \(G_e\) (resp. \(G_{i,e_i}\)) is the stabilizer of \(e\) (resp. \(e_i\)) in \(G\) (resp. \(G_i\)), we know that \[\mathrm{Aut}^G(O_e) \cong N(G_e)/G_e \cong \prod_{i=1}^d \left(N(G_{i,e_i})/G_{i,e_i}\right)\cong \prod_{i=1}^d \mathrm{Aut}^G(O_{e_i})\,,\] where \(N(G_e)\) is the normalizer of \(G_e\) in \(G\) and similarly for \(N(G_{i,e_i})\). Taking the symplectic form \(\sigma\) into account, the above product decomposition gives \[\mathrm{Aut}^G(O_e,[\sigma]) \subseteq \prod_{i=1}^d \mathrm{Aut}^{G_i}(O_{e_i},[\sigma_i]) \subseteq \prod_{i=1}^d \left(N(G_{i,e_i})/G_{i,e_i}\right)\,.\] By BK-nilpotent?, we have \[\left(N(G_{i,e_i})/G_{i,e_i}\right)^\circ \cong \mathbb{G}_m \ltimes U_i\] for some unipotent group \(U_i\). From the discussion before the lemma, we also know that \(\mathrm{Aut}^G(O_{e_i},[\sigma_i])\) contains at least one \(\mathbb{G}_m\), thus \[\mathrm{Aut}^{G_i}(O_{e_i},[\sigma_i])^\circ = \mathbb{G}_m \ltimes U'_i\] for some unipotent group \(U'_i\). Note that \(U'_i\) acts trivially on \(\sigma_i\) (as a unipotent group does not have any non-trivial one-dimensional representation), and the \(\mathbb{G}_m\) factor is dilating on \(\sigma_i\) with weight one. Now \[\prod_{i=1}^d \mathrm{Aut}^{G_i}(O_{e_i},[\sigma_i])^\circ \cong \mathbb{G}_m^d \ltimes U\] where \(U=\prod_{i=1}^d U'_i\) acts trivially on \(\sigma\), while the only subgroup of \(\mathbb{G}_m^d\) that dilates \(\sigma\) is the diagonal subgroup. Hence \[\mathrm{Aut}^G(O_e,[\sigma])^\circ \cong \mathbb{G}_m\ltimes U\] as desired. ◻

Proof. Let \(H=G\times \mathbb{G}_m\subseteq \mathrm{Aut}(\tilde{o} \in \widetilde{O}_e,[\sigma])\). Then \(Z(H)=\{1\}\times \mathbb{G}_m\), since \(G\) is semisimple. The \(Z(H)\)-action on \(\tilde{o} \in \widetilde{O}_e\) is induced by the scaling action on \(\mathfrak g\), which is good, and \[\mathrm{Aut}^H(\tilde{o} \in \widetilde{O}_e,[\sigma])^\circ\subseteq \mathrm{Aut}^G(\tilde{o} \in \widetilde{O}_e,[\sigma])^\circ \cong \mathbb{G}_m\ltimes U\] for some unipotent group \(U\) by Lemma 61, hence \(Z(H)\cong \mathbb{G}_m\) is a maximal torus of \(\mathrm{Aut}^H(\tilde{o} \in \widetilde{O}_e,[\sigma])\). This shows that \(H\) satisfies all the conditions of Lemma 60. Therefore, we conclude by Lemma 60 that the normalized volume minimizer is given by the Kollár component that corresponds to the \(Z(H)\)-action, i.e. the \(\mathbb{G}_m\) scaling action on \(\tilde{o} \in \widetilde{O}_e\).

Since \(\mathbb{C}[\overline{O}_e]\) is generated by its degree one elements (which are naturally identified with \(\mathfrak g\)), we know the \(\mathbb{G}_m\)-action on \(\overline{O}_e\setminus \{\overline{o}\}\) is free. As \(\widetilde{O}_e\to \overline{O}_e\) is isomorphic in codimension one (see BK-nilpotent?), the \(\mathbb{G}_m\)-action on \(\widetilde{O}_e\setminus \{\tilde{o}\}\) is free in codimension one. Thus the quotient is a variety (instead of a pair). Therefore, it is a K-semistable Fano variety by LX-Kol-comp-stab?. ◻

Proof. Assume the \(\mathbb{G}_m\)-action on \(\sigma\) has weight \(\ell>0\). It is enough to construct the Hamiltonian function for each \(\xi\in\mathfrak t=\operatorname{Lie}(\mathbb{T})\). Let \(\xi_Y\) denote the corresponding vector field on \(Y\). Since the \(\mathbb{T}\)-action is symplectic, it preserves \(\sigma\) and we have \(\mathcal{L}_{\xi_Y}\sigma=0\). Hence, by Cartan’s formula, \[\mathrm{d}(\iota_{\xi_Y}\sigma)=\mathcal{L}_{\xi_Y}\sigma-\iota_{\xi_Y}\mathrm{d}\sigma=0.\] Thus \(\alpha_\xi:=\iota_{\xi_Y}\sigma\) is a closed regular \(1\)-form.

Let \(\eta\) be the Euler vector field corresponding to the conical \(\mathbb{G}_m\)-action. Then \(\mathcal{L}_{\eta}\sigma = \ell\sigma\). Because the \(\mathbb{T}\)-action commutes with the conical \(\mathbb{G}_m\)-action, we have \([\eta,\xi_Y]=0\). Therefore \[\mathcal{L}_\eta\alpha_\xi = \mathcal{L}_\eta(\iota_{\xi_Y}\sigma) = \iota_{[\eta,\xi_Y]}\sigma+\iota_{\xi_Y}(\mathcal{L}_\eta\sigma) = \ell\,\iota_{\xi_Y}\sigma =\ell\alpha_\xi .\] On the other hand, applying Cartan’s formula to the closed \(1\)-form \(\alpha_\xi\), we get \[\mathcal{L}_\eta\alpha_\xi = \mathrm{d}(\iota_\eta \alpha_\xi)+\iota_\eta(\mathrm{d}\alpha_\xi) = \mathrm{d}(\iota_\eta\alpha_\xi).\] Since \(\ell>0\), combining the two identities gives \[\alpha_\xi = \mathrm{d}\left(\frac{1}{\ell}\iota_\eta\alpha_\xi\right) = \mathrm{d}\left(\frac{1}{\ell}\iota_\eta\iota_{\xi_Y}\sigma\right).\] Thus the Hamiltonian function for \(\xi\) is \[H_\xi = \frac{1}{\ell}\iota_\eta\iota_{\xi_Y}\sigma = \frac{1}{\ell}\sigma(\xi_Y,\eta),\] up to the usual sign convention for contractions. In particular, \[\mathrm{d}H_\xi=\iota_{\xi_Y}\sigma \, ,\] so the action is Hamiltonian. ◻

Proof. For \(\xi\in \operatorname{Lie}(\mathbb{T})\), let \(\xi_Y\) denote the corresponding vector field on \(Y\). Since the torus action under consideration is Hamiltonian, there is a Hamiltonian function \(H_\xi\) such that \(\mathrm{d}H_\xi=\iota_{\xi_Y}\sigma\). For two elements \(\xi,\eta\in \operatorname{Lie}(\mathbb{T})\), as the torus \(\mathbb{T}\) is abelian, the Hamiltonian functions Poisson commute: \[\{H_\xi,H_\eta\}=0 .\] But \(\{H_\xi,H_\eta\}= \sigma(\xi_Y,\eta_Y)\). Therefore, \(\sigma(\xi_Y,\eta_Y)=0\) for all \(\xi,\eta\in \operatorname{Lie}(\mathbb{T})\). Hence the tangent space to the \(\mathbb{T}\)-orbit at a general smooth point \(y\in Y\) is an isotropic subspace of the symplectic vector space \(T_yY\). Therefore, \(\dim (\mathbb{T})\le \frac{1}{2}\dim Y\). ◻

Proof. It is easy to see that the diagonal \(\mathbb{G}_m\)-action descends to an action \(\lambda\) on \(Y(A,0)\) and preserves the symplectic form \(\sigma_Y\) (with weight \(2\)).

Since \(\dim (Y(A,0))=2n\), by Lemmas 65 and 66, the maximal torus which acts on \((Y(A,0),\sigma_Y)\) by symplectomorphisms and commutes with \(\lambda\) is of dimension at most \(n\). So \(\mathbb{T}_Y\) is such a maximal torus. Therefore, the maximal torus which dilates \((Y(A,0),\sigma_Y)\) is of dimension \(n+1\), and \(\mathbb{T}'=\mathbb{G}_m\times \mathbb{T}_Y\) yields such a maximal torus, where the action of the first factor is \(\lambda\).

By Theorem 52, we know that the Reeb cone of \(\mathbb{T}'\) contains a minimizer of the normalized volume function for \(0\in Y(A,0)\). There is an involution \[\iota:\mathbb{C}^{2N}\longrightarrow \mathbb{C}^{2N}, \qquad \iota(z_i,w_i)=(w_i,z_i),\] which preserves \(\mu^{-1}(0)\) and \(\sigma_{\mathrm{st}}\), hence descends to a symplectic automorphism of \(Y(A,0)\). Note that \(\iota\) commutes with \(\lambda\) and \(\iota^{-1}\mathbb{T}_Y\iota=\mathbb{T}_Y\), and the conjugation action by \(\iota\) on \(\mathbb{T}_Y\) swaps the weights on \(z_i\) and \(w_i\). Let \(G\cong \mathbb{Z}/2\mathbb{Z}\ltimes \mathbb{T}'\) be the subgroup of \(\mathrm{Aut}(Y(A,0),[\sigma_Y])\) generated by \(\iota\) and \(\mathbb{T}'\). Then \(Z(G)=\mathbb{G}_m\), generated by the projection of the standard dilation on \(\mathbb{C}^{2N}\). By Lemma [l-center], we see that it gives the minimizer of \(Y(A,0)\). ◻

Proof. We may assume that \(V_0\) is also convex. Consider the vector field \(\widetilde{\xi}\) on \(X\times U_0\) given by \(\widetilde{\xi}(\mathbf{x},t) = (\xi_t,\partial_t)\). By Kau-group-action?, it is induced by a local group action on \(X\times U_0\). In other words, there exists an open neighborhood \(U\) of \(X\times U_0\times \{0\}\) in \(X\times U_0\times U_0\) and a morphism \(U\to X\times U_0\) of the form \[(\mathbf{x},s,t)\mapsto f_t(\mathbf{x},s) = (\widetilde{\varphi}_t(\mathbf{x},s),s+t)\] such that \(f_0=\mathrm{id}\) and \(f_t\circ f_{t'} = f_{t+t'}\) whenever both sides are well-defined. Because the vector field \(\xi_t\) fixes \(x\), we have \(f_t(x,s)=(x,s+t)\) for any \(s\in U_0\) and \(0<|t|\ll 1\) (depending on \(s\)). By the compactness of \(V_0\), we can then find some \(0<\varepsilon\ll 1\) and an analytic open neighborhood \(U_x\) of \(x\in X\) such that \(f_t(\mathbf{x},s)\) is defined for all \(\mathbf{x}\in U_x\), \(s\in V_0\) and \(|t|<\varepsilon\). This in turn implies that the composition \[f_{t/k}\circ \dots \circ f_{t/k}{\;\;\;(k times)}\] is well defined on some open neighborhood \(U'\) of \((x,0)\in X\times U_0\) for some sufficiently large integer \(k\) and for all \(t\in V_0\). It is an analytic continuation of \(f\) to \(U'\times V_0\) since \(f\) defines a group action. Therefore, we may assume from the beginning that \(U'\times V_0\subseteq U\). Then \[\varphi_t (\mathbf{x}) := \widetilde{\varphi}_t (\mathbf{x},0), \quad t\in V_0\] defines an automorphism of the analytic germ \(x\in X\) and satisfies the conditions of the lemma by construction. ◻

Proof. Define a stratification \[X_0\supseteq X_1\supseteq \dots\supseteq X_m\] of \(X\) inductively by \(X_0:=X\) and \(X_i:=\mathrm{Sing}(X_{i-1})\) until the last stratum becomes smooth. This stratification is preserved by any automorphism of \(X\) and hence also by all vector fields. In particular, the ideal sheaf of \(X_i\) is preserved by the derivation \(\partial_\xi\) corresponding to \(\xi\), thus \(\partial_\xi\) also defines a derivation on \(\mathcal{O}_{X_i}\) and hence a vector field \(\xi_i\) on each \(X_i\). By Kaledin-sym? (and the proof of Kaledin-sym? which gives the precise form of the stratification), every \(X_i\) is a Poisson subscheme of \(X\) whose induced Poisson structure is given by a symplectic form \(\sigma_i\). By construction, the smallest stratum \(X_m\) is smooth.

If \(x_1,\dots,x_r \in \mathcal{O}_{X_m,x}\) are local coordinates at \(x\in X_m\), and \(\eta_i:=H_{x_i}\) (\(i=1,\dots,r\)) are the corresponding Hamiltonian vector field on \(X_m\), then \(\eta_1,\dots,\eta_r\) spans \(T_x X_m\). In particular, we can choose some linear combination \(\bar{f}=\sum_{i=1}^r a_i x_i \in \mathcal{O}_{X_m,x}\) (\(a_i\in \mathbb{C}\)) of the \(x_i\) such that the vector field \[\xi_m-H_{\bar{f}} = \xi_m - \sum_{i=1}^r a_i\eta_i\] on \(X_m\) vanishes at \(x\), which implies \[\partial_{\xi_m}(\bar{g})-\{\bar{f},\bar{g}\}\in \mathfrak{m}_x \mathcal{O}_{X_m,x}\;\;for any \bar{g}\in \mathfrak{m}_x\, .\] Therefore, if \(f\in\mathcal{O}_{X,x}\) is any lift of \(\bar{f}\), then the above condition yields \[\partial_\xi(g) - \{f,g\}\in \mathfrak{m}_x\] for any \(g\in \mathfrak{m}_x\). In other words, \(\xi-H_f\) fixes \(x\in X\). ◻

Proof of Theorem 69. The basic ingredients are Moser’s trick Moser-trick? and the exactness of the symplectic forms proved by Kaledin Kaledin-sym?, though the singularity introduces some mild subtleties.

For \(\tau\in\mathbb{C}\), let \(\sigma_\tau := (1-\tau)\sigma_0 + \tau \sigma_1\). Note that \(\sigma_\tau\) is a symplectic form if and only if the induced section \(\sigma_\tau^{n/2}\in H^0(X,\omega_X)\) is non-vanishing at \(x\), which is the case when \(\tau=0,1\). Thus, there are only finitely many values of \(\tau\in\mathbb{C}\) for which \(\sigma_\tau\) is not symplectic. Hence to prove the theorem, we may assume that \(\sigma_t\) is symplectic for all \(t\in [0,1]\); this is because for general \(\tau\in \mathbb{C}\) and \(\sigma'=\sigma_\tau\), any convex combination of \(\sigma_j\) (\(j=0,1\)) and \(\sigma'\) is symplectic, and it is enough to find an automorphism that takes \(\sigma'\) to \(\sigma_i\) for each \(i\).

By Kaledin-sym?, possibly after shrinking \(X\) around \(x\), there exists a reflexive \(1\)-form \(\alpha\) on \(X\) such that \[\label{eq:symp32form32exact} \sigma_1 - \sigma_0 = \mathrm{d}\alpha.\tag{19}\] Consider the vector field \(\xi'_t\) on \(X\) defined by \(\alpha = \iota_{\xi'_t}(\sigma_t)\). Replacing \(\alpha\) by \(\alpha-\mathrm{d}f\) for \(f\in \mathcal{O}_{X,x}\) preserves the condition 19 , while the corresponding vector field changes to \(\xi'_t-H_f\), where \(H_f\) is the Hamiltonian vector field with respect to \(\sigma_t\). Therefore, by Lemma 71, we can choose an algebraic family of reflexive \(1\)-forms \(\alpha_t\) on \(X\) such that \[\mathrm{d}\alpha_t = \sigma_1-\sigma_0 = \frac{\mathrm{d}\sigma_{t}}{\mathrm{d}t},\] and the vector field \(\xi_t\) defined by \(\alpha_t = \iota_{\xi_t}(\sigma_t)\) fixes \(x\) for all \(t\). As \(x\) is fixed by all vector fields \(\xi_t\), we know from Lemma 70 that there exists a one-parameter family of automorphisms \(\varphi_t\in \mathrm{Aut}(X^{\mathrm{an}},x)\), \(t\in [0,1]\), such that \(\varphi_0=\mathrm{id}\) and \(\frac{\mathrm{d}\varphi_t}{\mathrm{d}t} = \xi_t\). We then have \[\frac{\mathrm{d}}{\mathrm{d}t} \varphi_t^*\sigma_0 = \mathcal{L}_{\xi_t} (\varphi_t^*\sigma_0) = \mathrm{d}\big(\iota_{\xi_t}(\varphi_t^*\sigma_0)\big),\] where the second equality is by Cartan’s formula. Since we also have \(\frac{\mathrm{d}\sigma_t}{\mathrm{d}t} = \mathrm{d}\big(\iota_{\xi_t}(\sigma_t)\big)\) and \(\varphi^*_0 \sigma_0 = \sigma_0\), we must have \(\varphi^*_t \sigma_0 = \sigma_t\) for all \(t\in [0,1]\). In particular, \(\varphi_1^*\sigma_0 = \sigma_1\). ◻

Proof. The metric \(j_t^*h\) develops two kinds of singularities, due to the facts that \(j_t\) need not be injective as a map of vector bundles (along \(S\cup F\)) and that \(h_{t,\varepsilon}\) (hence \(h\), too) may be (mildly) singular along \(f^{-1}(B)\). We fix a smooth hermitian metric \(h_L\) (resp. \(h_{V_t}\)) on \(L\) (resp. \(V_t\)), write \[j_t^*h=h_Le^{\psi}\] and consider the function \(\|j_t\|^2\) which is the squared norm of the morphism \(j_t\) with respect to \(h_{V_t}\) and \(h_L\). In other words, given any \(e\in L\setminus \{0\}\), we have \(\|j_t\|^2=\frac{|j_t(e)|^2_{h_{V_t}}}{|e|^2_{h_L}}\). We also consider the function \(\rho:=\log \frac{|j_t(e)|^2_{h }}{|j_t(e)|^2_{h_{V_t}}}\). All in all, we have \[\psi=\log \|j_t\|^2+\rho\] and \(\rho \in L^{\infty}(X)\) thanks to 33 .

Let \(\mu: Z\to \widetilde{Y}\) be a log resolution of the ideal of \(S\cup f^{-1}(B+F)\), with exceptional divisor \(\Gamma=\sum_{k\in K} \Gamma_k\). It is elementary to check that we have a decomposition \[\pi^*\log \|j_t\|^2=\sum_{k\in K} \lambda_k \log |s_{\Gamma_k}|^2+\sum_{j\in J} \nu_j\log |s_{F'_j}|^2+ \sigma\] where \(\lambda_k \ge 0\) (resp. \(\nu_j \ge 0\)), \(s_{\Gamma_k}\) (resp. \(s_{F_j'}\)) is “the” canonical section of \(\mathcal{O}_Z(\Gamma_k)\) (resp. \(\mathcal{O}_Z(F_j')\) where \(F_j'\) is the strict transform of \(f^{-1}(F_j)=f^*F_j\)) , measured with respect to some fixed smooth hermitian metric \(h_{\Gamma_k}\) (resp. \(h_{F_j'}\)), and \(\sigma \in \mathcal{C}^{\infty}(Z)\). In particular, we find on the complement of \(\Gamma\cup F'\) in \(Z\): \[\label{id} \mu^*\Theta(L, j_t^*h)=\mu^*\Theta(L,h_L)+\sum_{k\in K}\lambda_k\Theta(\Gamma_k, h_{\Gamma_k})+\sum_{j\in J}\nu_j\Theta(F_j', h_{F_j'})+dd^c(\mu^*\rho+\sigma).\tag{39}\] Let \((\chi_\delta)_\delta\) be the family of cut-off functions for \(\Gamma \cup F'\subset Z\) considered e.g. in GP-conic?. It can be interchangeably viewed as a function on \(Z\) or \(\widetilde{Y}\). It has the property that \(\pm dd^c \chi_\delta \le \omega_\Gamma\) where \(\omega_\Gamma\) is a Kähler metric on \(Z\setminus \Gamma\) with Poincaré type singularities along \(\Gamma\). Also, recall that for \(t,\varepsilon>0\) fixed, the metric \(f^*\omega_{t,\varepsilon}\) is dominated by a Kähler metric on \(\widetilde{Y}\). In particular, an integration by part yields \[\label{int32zero} \left| \int_{Z} \chi_\delta \, dd^c(\mu^*\rho+\sigma) \wedge \mu^*f^*\omega_{t,\varepsilon}^{n-1} \right| \le C_{t,\varepsilon}\cdot \|\mu^*\rho+\sigma\|_{L^{\infty}(Z)}\cdot \int_{\mathrm{Supp}(d\chi_\delta)} \omega_\Gamma^n\tag{40}\] and the latter goes to zero when \(\delta \to 0\).

Now, since \(\omega_{t,\varepsilon}\) has bounded potentials, we have \[c_1(L) \cdot f^*\alpha_t^{n-1}=\int_{\widetilde{Y}} \Theta(L,h_L) \wedge f^*\omega_{t,\varepsilon}^{n-1} = \lim_{\delta \to 0} \int_Z \chi_\delta \mu^*\Theta(L,h_L) \wedge \mu^*f^*\omega_{t,\varepsilon}^{n-1}\] and by 39 and 40 , we infer that \(c_1(L) \cdot f^*\alpha_t^{n-1}\) is computed by the quantity \[\lim_{\delta \to 0} \int_{\widetilde{Y}}\chi_\delta \, \Theta(L, j_t^*h)\wedge f^*\omega_{t,\varepsilon}^{n-1}-\sum_{k\in K} \lambda_k \underbrace{c_1(\Gamma_k) \cdot \mu^*f^*\alpha_t^{n-1}}_{=0}-\mathrm{deg}(f) \sum_{j\in J} \nu_j c_1(F_j) \cdot \alpha_t^{n-1},\] where the vanishing of the intersection numbers above holds since the \(\Gamma_k\)’s are \(\mu\)-exceptional. The lemma now follows from the fact that \(c_1(F_j) \cdot \alpha_t^{n-1}\) is a non-negative \(O(t)\) term since \(F_j\) is \(\pi\)-exceptional and \(\alpha_t=\pi^*\alpha+O(t)\). ◻

Proof. The Chern curvature form of the metric \(\tilde{h}_{t,\varepsilon}\) on \(\widetilde{E}_t\) is given by \[\Theta(\widetilde{E}_t,\tilde{h}_{t,\varepsilon})= \begin{pmatrix} \beta_{t,\varepsilon}\wedge \beta_{t,\varepsilon}^* & -\nabla^{1,0}_{h_{t,\varepsilon}}\beta_{t,\varepsilon}\\ -\bar \partial \beta_{t,\varepsilon}^* & \Theta(\Omega_{(\widetilde{X}, \widetilde{\Delta}, f)}^1, h_{t,\varepsilon}) +\beta_{t,\varepsilon}^*\wedge \beta_{t,\varepsilon} \end{pmatrix}\] away from \(f^{-1}(B)\). Arguing as in DGP-Q-Fano-decomp? one infers that, up to replacing \(\beta_{t,\varepsilon}\) by \(\frac{1+t}{\sqrt{n+1}}\beta_{t,\varepsilon}\)³, one has \[\label{curv} \Theta(\widetilde{E}_t,\tilde{h}_{t,\varepsilon})\wedge f^*\omega_{t,\varepsilon}^{n-1}=-\frac{1}{n+1} f^*\omega_{t,\varepsilon}^{n} \otimes \mathrm{Id}_{\widetilde{E}_t}-f^*\omega_{t,\varepsilon}^{n} \otimes A_{t,\varepsilon}\tag{41}\] where \[A_{t,\varepsilon}=\sharp_{f^*\omega_{t,\varepsilon}}f^*\big[-t \omega_{\widetilde{X}}+dd^c(\psi_\varepsilon-\varphi_{t,\varepsilon})-\Theta_\varepsilon\big],\] see 31 . The rest also follows very closely DGP-Q-Fano-decomp? (see also GSS?), with the only difference being the appearance of the function \(\chi_\delta\) to cut off the singularities of \(f^*\omega_{t,\varepsilon}\). The key elementary result that we need is that if \(\alpha\) is a positive \((1,1)\)-form on \(\widetilde{X}\), then \[\label{lin32alg} \|\sharp_{\omega_{t,\varepsilon}} \alpha\|_{\omega_{t,\varepsilon}}\, \omega_{t,\varepsilon}^n \le \mathrm{tr}(\sharp_{\omega_{t,\varepsilon}} \alpha) \, \omega_{t,\varepsilon}^n =n\alpha \wedge \omega_{t,\varepsilon}^{n-1}.\tag{42}\] There are \(4\) terms in the expansion of \(A_{t,\varepsilon}\), say \(A_1, \ldots, A_4\). Clearly, one has \[\|A_1+A_2\|\, f^*\omega_{t,\varepsilon}^n \le Ct \cdot f^*(\omega_{\widetilde{X}}\wedge \omega_{t,\varepsilon}^{n-1})\] Next, the argument given in DGP-Q-Fano-decomp? shows that \[\limsup_{\varepsilon\to 0} \int_{\widetilde{Y}} \|A_3\| f^*\omega_{t,\varepsilon}^n=0.\] As for \(A_4\), 34 and 42 combined with the Lebesgue dominated convergence theorem show that for fixed \(\varepsilon>0\), we have \[\lim_{\delta \to 0} \int_{\widetilde{Y}} \chi_\delta \|A_4\| f^*\omega_{t,\varepsilon}^n = \int_{\widetilde{Y}} \|A_4\| f^*\omega_{t,\varepsilon}^n.\] Then we write \(\Theta_\varepsilon=\sum_{j\in J} c_j (\Theta_{\varepsilon, j}+g_\varepsilon\omega_{\widetilde{X}})-\big(\sum_{j\in J} c_j\big) g_\varepsilon\omega_{\widetilde{X}}\) and use 42 to see that \[\int_{\widetilde{Y}} \|A_4\|f^*\omega_{t,\varepsilon}^n \le n\cdot \mathrm{deg}(f) \left(\sum_{j\in J}|c_j| (c_1(F_j) \cdot \alpha_t^{n-1})+\Big| \sum_{j\in J} c_j \Big| \int_{\widetilde{X}} g_\varepsilon\omega_{\widetilde{X}} \wedge \omega_{t,\varepsilon}^{n-1}\right)\] On the RHS, the first summand is independent of \(\varepsilon\) and goes to zero as \(t\to 0\) while the second summand goes to zero when \(\varepsilon\to 0\) and \(t>0\) is fixed thanks to 35 and Lebesgue’s dominated convergence theorem. Putting all the above estimates together yields \[\lim_{t\to 0} \lim_{\varepsilon\to 0} \lim_{\delta \to 0} \int_{\widetilde{Y}} \chi_\delta \, \| A_{t,\varepsilon}\| \, f^*\omega_{t,\varepsilon}^{n} =0.\] Given the definition of \((V_t,h)\) (see 37 ) and the fact that \(F\) is \(\pi\)-exceptional, we deduce from 41 and the above identity that \[\limsup_{t\to 0} \limsup_{\varepsilon\to 0} \limsup_{\delta \to 0} \int_{\widetilde{Y}} \chi_\delta \, \mathrm{tr}(\mathrm{pr}_L \Theta(V_t, h)|_L) \wedge f^*\omega_{t,\varepsilon}^{n-1}\le-\frac{r}{n+1}\cdot \deg(f) \cdot \alpha^n,\] hence the lemma. ◻

Proof of Theorem 72. Let us first assume that \((X,\Delta)\) is uniformly K-stable. Thanks to LTW-YTD?, there exists a Kähler-Einstein metric \(\omega\) on \((X,\Delta)\). Therefore, one can use the approximations \(\omega_{t,\varepsilon}\) of \(\omega\) on \(\widetilde{X}\) introduced in Section 9.4 to compute the slopes of subsheaves \(\widetilde{F} \subset \widetilde{E}\) as explained in Section 9.5. More precisely, one combines Lemma 73, Lemma 74 and 38 and infers that \(\widetilde{E}\) is semistable with respect to \(f^*\pi^*\alpha\). Thanks to 26 , this implies that \(E\) is semistable with respect to \(g^*\alpha\). This proves the theorem in the uniformly K-stable case.

If \((X,\Delta)\) is merely K-semistable, pick \(H\in |-m(K_X+\Delta)|\) general for \(m\) large and divisible and consider the log Fano pair \((X,\Delta_\varepsilon:=\Delta+\varepsilon H)\) for \(\varepsilon\in \mathbb{Q}_{>0}\) small. Let \(h_\varepsilon:Y_\varepsilon\to Y\) be an adapted cover for the pair \((Y,\varepsilon g^* H)\) and denote by \(g_\varepsilon: Y_\varepsilon\to X\) the induced map. We have a natural injection \[\label{inj32form} h_\varepsilon^{[*]}\Omega_{(X,\Delta, g)}^{[1]}\simeq \Omega_{(X,\Delta,g_\varepsilon)}^{[1]}\subset \Omega_{(X,\Delta_\varepsilon,g_\varepsilon)}^{[1]}.\tag{43}\] Next, let \(E_\varepsilon\) be the extension of \(\mathcal{O}_{Y_\varepsilon}\) by \(\Omega_{(X,\Delta_\varepsilon, g_\varepsilon)}^{[1]}\) with extension class induced by \(-(K_X+\Delta)\) under the map \(\mathrm{Pic}(X)_{\mathbb{Q}}\to H^1(Y_\varepsilon, \Omega_{(X,\Delta_\varepsilon, g_\varepsilon)}^{[1]})\), see 25 . It is easy to check directly using the transition maps 23 that 43 induces an injective map \[\label{inj32ext} h_\varepsilon^{[*]}E\subset E_\varepsilon,\tag{44}\] where \(E:=E_{(X,\Delta,g)}\) is the canonical extension of \((X,\Delta, g)\). Moreover, since \(K_X+\Delta_\varepsilon\) is proportional to \(K_X+\Delta\), \(E_\varepsilon\) is abstractly isomorphic to the canonical extension of \((X,\Delta_\varepsilon, g_\varepsilon)\), i.e. the extension of \(\mathcal{O}_{Y_\varepsilon}\) by \(\Omega_{(X,\Delta_\varepsilon, g_\varepsilon)}^{[1]}\) with extension class induced by \(-(K_X+\Delta_\varepsilon)\). Thanks to LXZ-HRFG? (the same proof works verbatim when \(t=0\)), the pair \((X,\Delta_\varepsilon)\) is uniformly K-stable for any \(\varepsilon>0\) small enough. Let \(F\subset E\) be a coherent subsheaf of rank \(r\), which induces a subsheaf \(h_\varepsilon^{[*]}F\subset E_\varepsilon\) via 44 . By the previous case, \(E_\varepsilon\) is semistable with respect to \(g_\varepsilon^*\alpha\) hence we get \[\begin{align} \frac{\deg(h_\varepsilon)}{r}(c_1(F) \cdot g^*\alpha^{n-1})&=& \frac{1}{r} (c_1(h_\varepsilon^{[*]}F) \cdot h_\varepsilon^*g^*\alpha^{n-1})\\ & \le& \frac{1}{n+1}(c_1(E_\varepsilon) \cdot g_\varepsilon^*\alpha^{n-1})\\ &=&-\frac{1-\varepsilon}{n+1} \cdot \deg(h_\varepsilon) \cdot g^*\alpha^n. \end{align}\] Dividing both sides by \(\deg(h_\varepsilon)\) and letting \(\varepsilon\) go to zero yields the desired slope semistability of \(E\) with respect to \(g^*\alpha\). ◻